On 2-Holonomy

Hossein AbbaspourUniversite de Nantes

Friedrich WagemannUniversite de Nantes

May 3 2019

Abstract

We construct a cycle in higher Hochschild homology associated to the2-dimensional torus which represents 2-holonomy of a non-abelian gerbein the same way as the ordinary holonomy of a principal G-bundle givesrise to a cycle in ordinary Hochschild homology This is done using theconnection 1-form of Baez-Schreiber

A crucial ingredient in our work is the possibility to arrange that inthe structure crossed module micro h rarr g of the principal 2-bundle the Liealgebra h is abelian up to equivalence of crossed modules

Keywords holonomy of a principal 2-bundle higher Hochschild homologycrossed modules of Lie algebras connection 1-form on loop space

Mathematics Subject Classifications (2010) 53C08 (primary) 17B5553C05 53C29 55R40 (secondary)

Contents

1 Strict Lie 2-algebras and crossed modules 611 Strict 2-vector spaces 612 Strict Lie 2-algebras and crossed modules 713 Semi-strict Lie 2-algebras and 2-term Linfin-algebras 914 Classification of semi-strict Lie 2-algebras 1115 The construction of an abelian representative 13

2 Crossed modules of Lie groups 15

3 Principal 2-bundles and gerbes 1631 Definition 1732 Connection data 18

4 Linfin-valued differential forms 20

1

5 Path space and the connection 1-form associated to a principal2-bundle 2251 Path space as a Frechet manifold 2252 The connection 1-form of Baez-Schreiber 22

6 The holonomy cycle associated to a principal 2-bundle 24

7 Proof of the main theorem 27

8 Explicit expression for the holonomy cycle 29

Introduction

The notion of a principal 2-bundle grew out of the notion of a non-abeliangerbe as defined by Giraud [Gi71] Breen [Bre94] and further studied by Breenand Messing [BreMe] on the one hand and Laurent-Gengoux Stienon and Xu[LSX10] on the other hand

Principal 2-bundles in the narrow sense have been studied in [BaSc04][Bar04] [GiSt08] [MaPi07] [ScWa08a] [ScWa08b] [Wo09] and [CLS10] (Thislist is by no means exhaustive) We will sketch the different approaches andexplain our point of view namely we choose a framework at the intersectionof gerbe theory and theory of principal 2-bundles The structure 2-group of aprincipal 2-bundle is in our framework a strict 2-group (we refrain from consid-ering more general structure groups like coherent 2-groups) and its Lie algebraa strict Lie 2-algebra opening the way to using all information about strict Lie2-algebras which we discuss in the first section

The first (non-gerbal) approach to principal 2-bundles is due to Bartels[Bar04] He defines 2-bundles by systematically categorifying spaces groupsand bundles Bartels writes down the necessary coherence relations for a lo-cally trivial principal 2-bundle with structure group a coherent 2-group Thiswork has then been taken up by Baez and Schreiber [BaSc04] in order to defineconnections for principal 2-bundles In parallel work Schreiber and Waldorf[ScWa08a] [ScWa08b] and Wockel [Wo09] also take up Bartels work in orderto define holonomy (Schreiber-Waldorf) or to pass to gauge groups (Wockel)Baez and Schreiber describe an approach using locally trivial 2-fibrations whosetypical fiber is a strict 2-group

Non-abelian gerbes and principal 2-bundles are two notions which are closebut have subtle differences The cocycle data of the two notions has beencompared in [BaSc04] section 214 and 22 Baez and Schreiber show thatunder certain conditions the description in terms of local data of a principal 2-bundle with 2-connection is equivalent to the cocycle description of a (possiblytwisted) non-abelian gerbe with vanishing fake curvature This constraint isalso shown to be sufficient for the existence of 2-holonomies ie the paralleltransport over surfaces

The approach of Schreiber and Waldorf [ScWa08a] [ScWa08b] is based on

2

so-called transport functors Schreiber and Waldorf push the equivalence be-tween categories of principal G-bundles with connection over M and transportfunctors from the thin fundamental groupoid of M to the classifying stack ofG to categorical dimension 2 These transport functors can then be describedin terms of differential forms ie for a trivial principal G-bundle these trans-port functors correspond to Ω1(M g) where g is the Lie algebra of G Theyshow similarly that 2-transport functors from the thin fundamental 2-groupoidcorrespond to pairs of differential forms A isin Ω1(M g) and B isin Ω1(M h) withvanishing fake curvature FA + micro(B) = 0 where micro h rarr g is the crossed moduleof Lie algebras corresponding to the strict Lie 2-group which comes into theproblem It is clear that this approach is based on the notion of holonomy

Wockel [Wo09] also takes up Bartelrsquos work In order to make them moreeasily accessible he formulates a principal 2-bundle over M in terms of spaceswith a group action A (semi-strict) principal 2-bundle over M is then a locallytrivial G-2-space The 2-group G is strict and so is the action functor but thelocal triviality requirement is not necessarily strict Wockel shows that semi-strict principal 2-bundles over M are classified by non-abelian Cech cohomology

The approach of Ginot and Stienon [GiSt08] is based on looking at a principalG-bundle as a generalized morphism (in the sense of Hilsum and Skandalis)from M to G both being considered as groupoids In the same way theyview principal 2-bundles as generalized morphisms from the manifold M (or ingeneral some stack represented by a Lie groupoid) to the 2-group G both beingviewed as 2-groupoids In this context they exhibit a link to gerbes (in theirincarnation as extensions of groupoids) and define characteristic classes

The particularity of Martins and Pickenrsquos approach [MaPi07] is that theyconsider special G-2-bundles For a strict 2-group G whose associated crossedmodule is micro H rarr G these bundles are obtained from a principal G-bundle Pon M The speciality requirement is that the principal G-2-bundle is given by anon-abelian cocycle (gij hijk) as below but with micro(hijk) = 1 in order to havea principal G-bundle P Using the language which we will introduce belowMartins and Picken suppose that the band of the gerbe (which is in generala principal Gmicro(H)-bundle) lifts to a principal G-bundle Martins and Pickendefine connections for these special G-2-bundles and 2-holonomy 2-functors

Chatterjee Lahiri and Sengupta [CLS10] use in the first place a referenceconnection 1-form A in order to take for a fixed G-principal bundle P rarr M onlyA-horizontal paths in the path space PAP they consider PAP is a G-principalbundle over the usual path space PM Then given a pair (AB) as abovethey construct a connection 1-form ω(AB) on PAP using Chen integrals Majorissues are reparametrization invariance and the curvature The authors switchto a categorical description motivated by their differential geometric study inthe end of the article

Let us summarize the different approaches in the following table

3

author(s) concept

Bartels principal 2-bundles withcoherent structure group

Baez-Schreiber global connection 1-formfor principal 2-bundles

Schreiber-Waldorf holonomy in terms oftransport functors

Wockel relation to non-abelianCech cohomology

Ginot-Stienon 2-bundles as Hilsum-Skandalisrsquogeneralized morphisms

Martin-Picken connections and holonomy forspecial principal 2-bundles

Chatterjee-Lahiri-Sengupta connections and holonomyusing A-horizontal pathsfor a reference 1-form A

Let us also mention the more recent paper by Nikolaus andWaldorf [NiWa11]where the equivalences between some of the above incarnations of non-abeliangerbes and principal 2-bundles are shown

The goal of our article is to construct a cycle in higher Hochschild homologywhich represents 2-holonomy of a non-abelian gerbe as described above in thesame way as the ordinary holonomy gives rise to a cycle in ordinary Hochschildhomology see [AbZe07] This is done using the connection 1-form of Baez-Schreiber [BaSc04] which we construct here from the band of the non-abeliangerbe

A crucial ingredient in our work is the possibility to arrange that in an ar-bitrary crossed module of Lie algebras micro h rarr g the Lie algebra h is abelianup to equivalence of crossed modules This is shown in Section 1 (see [Wa06])The possibility to have h abelian is used in order to obtain a commutativedifferential graded algebra Ωlowast = Ωlowast(MUh) whose higher Hochschild homol-ogy HHT

bull (ΩlowastΩlowast) associated to the 2-dimensional torus T houses the holonomy

cycle We do not know of any definition of higher Hochschild homology for arbi-trary differential graded algebras therefore we believe the reduction to abelianh to be crucial when working with possibly non-abelian gerbes Section 1 alsoprovides a fundamental result on strict Lie 2-algebras directly inspired from[BaCr04] namely we explicitly show that the two classifications of strict Lie 2-algebras in terms of skeletal models (of the associated semi-strict Lie 2-algebra)and in terms of the associated crossed modules coincide

Section 2 reports on crossed modules of Lie groups These play a minorrole in our study because the main ingredient for the connection data is thethe infinitesimal crossed module ie the Lie algebra crossed module Section 3gives the definition of principal 2-bundles with which we work It is taken fromWockelrsquos article [Wo09] together with restrictions from [BaSc04] In Section 4we discuss in general Linfin-valued differential forms on the manifold M based on

4

the article of Getzler [Ge09] We believe that this is the right generalization ofthe calculus of Lie algebra valued differential forms needed for ordinary principalG-bundles We find a curious 3-form term (see equation (2)) in the Maurer-Cartan equation for differential forms with values in a semi-strict Lie 2-algebrawhich also appears in [He08] In Section 5 we construct the connection 1-formA0 of Baez-Schreiber from the band of the non-abelian gerbe It is not soclear in [BaSc04] on which differential geometric object the construction of A0

is carried out and we believe that expressing it as the usual iterated integralconstruction on the band (which is an ordinary principal G-bundle ) is ofconceptual importance

Section 6 is the heart of our article and explains the mechanism to trans-form the flat connection A0 into a Hochschild cycle for the differential gradedalgebra CHlowast(Ω

lowastΩlowast) It lives therefore in the Hochschild homology of the al-gebra of Hochschild chains Section 7 recalls from [GTZ09] that ldquoHochschildof Hochschildrdquo-homology is the higher Hochschild homology associated to thetorus T2 We proceed with an explicit expression for (some terms arising in)the holonomy cycle in Section 8

In this article to we try to give a more conceptual approch to the holonomy ofgerbes rather than a computational one as it is introduced [GaRe02] and furtherdevelopped in [TWZ11] Another important difference is that our approachincludes non-abelian gerbes as well

The main theorem of the present article is the construction of the homologycycle representing the holonomy It should be thought of as a 0-cochain onmapping surface (torus in this particular case) space

Theorem 1 Consider a non-abelian principal 2-bundle with trivial band on amanifold M with a structure crossed module micro h rarr g such that the Lie algebrah is abelian Then the connection 1-form A0 of Baez-Schreiber gives rise to acycle P (A0) in the higher Hochschild homology HHT

bull (ΩlowastΩlowast) which corresponds

to the holonomy of the gerbe

As stated before we do not consider the condition that h is abelian as arestriction of generality because up to equivalence it may be achieved for anarbitrary crossed module

By construction the cycle P (A0) is not always trivial ie a boundarybecause it represents the holonomy Observe that for the crossed module id h rarr h we recover the result of [AbZe07] The triviality condition on the bandmay be understood as expressing that the construction is local The gluing of thelocally defined connection 1-forms of Baez and Schreiber to a global connection1-form (see [BaSc04]) should permit to glue our Hochschild cycles

Another subject of further research is to understand that the connection1-form A0 does not only lead to a higher Hochschild cycle with respect to the 2-dimensional torus but actually to higher Hochschild cycles with respect to anycompact topological surface In fact we believe that there is a way to recover

HHΣgbull for a connected compact surface Σg of genus g from HHT

bull

5

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

5 Path space and the connection 1-form associated to a principal2-bundle 2251 Path space as a Frechet manifold 2252 The connection 1-form of Baez-Schreiber 22

6 The holonomy cycle associated to a principal 2-bundle 24

7 Proof of the main theorem 27

8 Explicit expression for the holonomy cycle 29

Introduction

The notion of a principal 2-bundle grew out of the notion of a non-abeliangerbe as defined by Giraud [Gi71] Breen [Bre94] and further studied by Breenand Messing [BreMe] on the one hand and Laurent-Gengoux Stienon and Xu[LSX10] on the other hand

Principal 2-bundles in the narrow sense have been studied in [BaSc04][Bar04] [GiSt08] [MaPi07] [ScWa08a] [ScWa08b] [Wo09] and [CLS10] (Thislist is by no means exhaustive) We will sketch the different approaches andexplain our point of view namely we choose a framework at the intersectionof gerbe theory and theory of principal 2-bundles The structure 2-group of aprincipal 2-bundle is in our framework a strict 2-group (we refrain from consid-ering more general structure groups like coherent 2-groups) and its Lie algebraa strict Lie 2-algebra opening the way to using all information about strict Lie2-algebras which we discuss in the first section

The first (non-gerbal) approach to principal 2-bundles is due to Bartels[Bar04] He defines 2-bundles by systematically categorifying spaces groupsand bundles Bartels writes down the necessary coherence relations for a lo-cally trivial principal 2-bundle with structure group a coherent 2-group Thiswork has then been taken up by Baez and Schreiber [BaSc04] in order to defineconnections for principal 2-bundles In parallel work Schreiber and Waldorf[ScWa08a] [ScWa08b] and Wockel [Wo09] also take up Bartels work in orderto define holonomy (Schreiber-Waldorf) or to pass to gauge groups (Wockel)Baez and Schreiber describe an approach using locally trivial 2-fibrations whosetypical fiber is a strict 2-group

Non-abelian gerbes and principal 2-bundles are two notions which are closebut have subtle differences The cocycle data of the two notions has beencompared in [BaSc04] section 214 and 22 Baez and Schreiber show thatunder certain conditions the description in terms of local data of a principal 2-bundle with 2-connection is equivalent to the cocycle description of a (possiblytwisted) non-abelian gerbe with vanishing fake curvature This constraint isalso shown to be sufficient for the existence of 2-holonomies ie the paralleltransport over surfaces

The approach of Schreiber and Waldorf [ScWa08a] [ScWa08b] is based on

2

so-called transport functors Schreiber and Waldorf push the equivalence be-tween categories of principal G-bundles with connection over M and transportfunctors from the thin fundamental groupoid of M to the classifying stack ofG to categorical dimension 2 These transport functors can then be describedin terms of differential forms ie for a trivial principal G-bundle these trans-port functors correspond to Ω1(M g) where g is the Lie algebra of G Theyshow similarly that 2-transport functors from the thin fundamental 2-groupoidcorrespond to pairs of differential forms A isin Ω1(M g) and B isin Ω1(M h) withvanishing fake curvature FA + micro(B) = 0 where micro h rarr g is the crossed moduleof Lie algebras corresponding to the strict Lie 2-group which comes into theproblem It is clear that this approach is based on the notion of holonomy

Wockel [Wo09] also takes up Bartelrsquos work In order to make them moreeasily accessible he formulates a principal 2-bundle over M in terms of spaceswith a group action A (semi-strict) principal 2-bundle over M is then a locallytrivial G-2-space The 2-group G is strict and so is the action functor but thelocal triviality requirement is not necessarily strict Wockel shows that semi-strict principal 2-bundles over M are classified by non-abelian Cech cohomology

The approach of Ginot and Stienon [GiSt08] is based on looking at a principalG-bundle as a generalized morphism (in the sense of Hilsum and Skandalis)from M to G both being considered as groupoids In the same way theyview principal 2-bundles as generalized morphisms from the manifold M (or ingeneral some stack represented by a Lie groupoid) to the 2-group G both beingviewed as 2-groupoids In this context they exhibit a link to gerbes (in theirincarnation as extensions of groupoids) and define characteristic classes

The particularity of Martins and Pickenrsquos approach [MaPi07] is that theyconsider special G-2-bundles For a strict 2-group G whose associated crossedmodule is micro H rarr G these bundles are obtained from a principal G-bundle Pon M The speciality requirement is that the principal G-2-bundle is given by anon-abelian cocycle (gij hijk) as below but with micro(hijk) = 1 in order to havea principal G-bundle P Using the language which we will introduce belowMartins and Picken suppose that the band of the gerbe (which is in generala principal Gmicro(H)-bundle) lifts to a principal G-bundle Martins and Pickendefine connections for these special G-2-bundles and 2-holonomy 2-functors

Chatterjee Lahiri and Sengupta [CLS10] use in the first place a referenceconnection 1-form A in order to take for a fixed G-principal bundle P rarr M onlyA-horizontal paths in the path space PAP they consider PAP is a G-principalbundle over the usual path space PM Then given a pair (AB) as abovethey construct a connection 1-form ω(AB) on PAP using Chen integrals Majorissues are reparametrization invariance and the curvature The authors switchto a categorical description motivated by their differential geometric study inthe end of the article

Let us summarize the different approaches in the following table

3

author(s) concept

Bartels principal 2-bundles withcoherent structure group

Baez-Schreiber global connection 1-formfor principal 2-bundles

Schreiber-Waldorf holonomy in terms oftransport functors

Wockel relation to non-abelianCech cohomology

Ginot-Stienon 2-bundles as Hilsum-Skandalisrsquogeneralized morphisms

Martin-Picken connections and holonomy forspecial principal 2-bundles

Chatterjee-Lahiri-Sengupta connections and holonomyusing A-horizontal pathsfor a reference 1-form A

Let us also mention the more recent paper by Nikolaus andWaldorf [NiWa11]where the equivalences between some of the above incarnations of non-abeliangerbes and principal 2-bundles are shown

The goal of our article is to construct a cycle in higher Hochschild homologywhich represents 2-holonomy of a non-abelian gerbe as described above in thesame way as the ordinary holonomy gives rise to a cycle in ordinary Hochschildhomology see [AbZe07] This is done using the connection 1-form of Baez-Schreiber [BaSc04] which we construct here from the band of the non-abeliangerbe

A crucial ingredient in our work is the possibility to arrange that in an ar-bitrary crossed module of Lie algebras micro h rarr g the Lie algebra h is abelianup to equivalence of crossed modules This is shown in Section 1 (see [Wa06])The possibility to have h abelian is used in order to obtain a commutativedifferential graded algebra Ωlowast = Ωlowast(MUh) whose higher Hochschild homol-ogy HHT

bull (ΩlowastΩlowast) associated to the 2-dimensional torus T houses the holonomy

cycle We do not know of any definition of higher Hochschild homology for arbi-trary differential graded algebras therefore we believe the reduction to abelianh to be crucial when working with possibly non-abelian gerbes Section 1 alsoprovides a fundamental result on strict Lie 2-algebras directly inspired from[BaCr04] namely we explicitly show that the two classifications of strict Lie 2-algebras in terms of skeletal models (of the associated semi-strict Lie 2-algebra)and in terms of the associated crossed modules coincide

Section 2 reports on crossed modules of Lie groups These play a minorrole in our study because the main ingredient for the connection data is thethe infinitesimal crossed module ie the Lie algebra crossed module Section 3gives the definition of principal 2-bundles with which we work It is taken fromWockelrsquos article [Wo09] together with restrictions from [BaSc04] In Section 4we discuss in general Linfin-valued differential forms on the manifold M based on

4

the article of Getzler [Ge09] We believe that this is the right generalization ofthe calculus of Lie algebra valued differential forms needed for ordinary principalG-bundles We find a curious 3-form term (see equation (2)) in the Maurer-Cartan equation for differential forms with values in a semi-strict Lie 2-algebrawhich also appears in [He08] In Section 5 we construct the connection 1-formA0 of Baez-Schreiber from the band of the non-abelian gerbe It is not soclear in [BaSc04] on which differential geometric object the construction of A0

is carried out and we believe that expressing it as the usual iterated integralconstruction on the band (which is an ordinary principal G-bundle ) is ofconceptual importance

Section 6 is the heart of our article and explains the mechanism to trans-form the flat connection A0 into a Hochschild cycle for the differential gradedalgebra CHlowast(Ω

lowastΩlowast) It lives therefore in the Hochschild homology of the al-gebra of Hochschild chains Section 7 recalls from [GTZ09] that ldquoHochschildof Hochschildrdquo-homology is the higher Hochschild homology associated to thetorus T2 We proceed with an explicit expression for (some terms arising in)the holonomy cycle in Section 8

In this article to we try to give a more conceptual approch to the holonomy ofgerbes rather than a computational one as it is introduced [GaRe02] and furtherdevelopped in [TWZ11] Another important difference is that our approachincludes non-abelian gerbes as well

The main theorem of the present article is the construction of the homologycycle representing the holonomy It should be thought of as a 0-cochain onmapping surface (torus in this particular case) space

Theorem 1 Consider a non-abelian principal 2-bundle with trivial band on amanifold M with a structure crossed module micro h rarr g such that the Lie algebrah is abelian Then the connection 1-form A0 of Baez-Schreiber gives rise to acycle P (A0) in the higher Hochschild homology HHT

bull (ΩlowastΩlowast) which corresponds

to the holonomy of the gerbe

As stated before we do not consider the condition that h is abelian as arestriction of generality because up to equivalence it may be achieved for anarbitrary crossed module

By construction the cycle P (A0) is not always trivial ie a boundarybecause it represents the holonomy Observe that for the crossed module id h rarr h we recover the result of [AbZe07] The triviality condition on the bandmay be understood as expressing that the construction is local The gluing of thelocally defined connection 1-forms of Baez and Schreiber to a global connection1-form (see [BaSc04]) should permit to glue our Hochschild cycles

Another subject of further research is to understand that the connection1-form A0 does not only lead to a higher Hochschild cycle with respect to the 2-dimensional torus but actually to higher Hochschild cycles with respect to anycompact topological surface In fact we believe that there is a way to recover

HHΣgbull for a connected compact surface Σg of genus g from HHT

bull

5

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

so-called transport functors Schreiber and Waldorf push the equivalence be-tween categories of principal G-bundles with connection over M and transportfunctors from the thin fundamental groupoid of M to the classifying stack ofG to categorical dimension 2 These transport functors can then be describedin terms of differential forms ie for a trivial principal G-bundle these trans-port functors correspond to Ω1(M g) where g is the Lie algebra of G Theyshow similarly that 2-transport functors from the thin fundamental 2-groupoidcorrespond to pairs of differential forms A isin Ω1(M g) and B isin Ω1(M h) withvanishing fake curvature FA + micro(B) = 0 where micro h rarr g is the crossed moduleof Lie algebras corresponding to the strict Lie 2-group which comes into theproblem It is clear that this approach is based on the notion of holonomy

Wockel [Wo09] also takes up Bartelrsquos work In order to make them moreeasily accessible he formulates a principal 2-bundle over M in terms of spaceswith a group action A (semi-strict) principal 2-bundle over M is then a locallytrivial G-2-space The 2-group G is strict and so is the action functor but thelocal triviality requirement is not necessarily strict Wockel shows that semi-strict principal 2-bundles over M are classified by non-abelian Cech cohomology

The approach of Ginot and Stienon [GiSt08] is based on looking at a principalG-bundle as a generalized morphism (in the sense of Hilsum and Skandalis)from M to G both being considered as groupoids In the same way theyview principal 2-bundles as generalized morphisms from the manifold M (or ingeneral some stack represented by a Lie groupoid) to the 2-group G both beingviewed as 2-groupoids In this context they exhibit a link to gerbes (in theirincarnation as extensions of groupoids) and define characteristic classes

The particularity of Martins and Pickenrsquos approach [MaPi07] is that theyconsider special G-2-bundles For a strict 2-group G whose associated crossedmodule is micro H rarr G these bundles are obtained from a principal G-bundle Pon M The speciality requirement is that the principal G-2-bundle is given by anon-abelian cocycle (gij hijk) as below but with micro(hijk) = 1 in order to havea principal G-bundle P Using the language which we will introduce belowMartins and Picken suppose that the band of the gerbe (which is in generala principal Gmicro(H)-bundle) lifts to a principal G-bundle Martins and Pickendefine connections for these special G-2-bundles and 2-holonomy 2-functors

Chatterjee Lahiri and Sengupta [CLS10] use in the first place a referenceconnection 1-form A in order to take for a fixed G-principal bundle P rarr M onlyA-horizontal paths in the path space PAP they consider PAP is a G-principalbundle over the usual path space PM Then given a pair (AB) as abovethey construct a connection 1-form ω(AB) on PAP using Chen integrals Majorissues are reparametrization invariance and the curvature The authors switchto a categorical description motivated by their differential geometric study inthe end of the article

Let us summarize the different approaches in the following table

3

author(s) concept

Bartels principal 2-bundles withcoherent structure group

Baez-Schreiber global connection 1-formfor principal 2-bundles

Schreiber-Waldorf holonomy in terms oftransport functors

Wockel relation to non-abelianCech cohomology

Ginot-Stienon 2-bundles as Hilsum-Skandalisrsquogeneralized morphisms

Martin-Picken connections and holonomy forspecial principal 2-bundles

Chatterjee-Lahiri-Sengupta connections and holonomyusing A-horizontal pathsfor a reference 1-form A

Let us also mention the more recent paper by Nikolaus andWaldorf [NiWa11]where the equivalences between some of the above incarnations of non-abeliangerbes and principal 2-bundles are shown

The goal of our article is to construct a cycle in higher Hochschild homologywhich represents 2-holonomy of a non-abelian gerbe as described above in thesame way as the ordinary holonomy gives rise to a cycle in ordinary Hochschildhomology see [AbZe07] This is done using the connection 1-form of Baez-Schreiber [BaSc04] which we construct here from the band of the non-abeliangerbe

A crucial ingredient in our work is the possibility to arrange that in an ar-bitrary crossed module of Lie algebras micro h rarr g the Lie algebra h is abelianup to equivalence of crossed modules This is shown in Section 1 (see [Wa06])The possibility to have h abelian is used in order to obtain a commutativedifferential graded algebra Ωlowast = Ωlowast(MUh) whose higher Hochschild homol-ogy HHT

bull (ΩlowastΩlowast) associated to the 2-dimensional torus T houses the holonomy

cycle We do not know of any definition of higher Hochschild homology for arbi-trary differential graded algebras therefore we believe the reduction to abelianh to be crucial when working with possibly non-abelian gerbes Section 1 alsoprovides a fundamental result on strict Lie 2-algebras directly inspired from[BaCr04] namely we explicitly show that the two classifications of strict Lie 2-algebras in terms of skeletal models (of the associated semi-strict Lie 2-algebra)and in terms of the associated crossed modules coincide

Section 2 reports on crossed modules of Lie groups These play a minorrole in our study because the main ingredient for the connection data is thethe infinitesimal crossed module ie the Lie algebra crossed module Section 3gives the definition of principal 2-bundles with which we work It is taken fromWockelrsquos article [Wo09] together with restrictions from [BaSc04] In Section 4we discuss in general Linfin-valued differential forms on the manifold M based on

4

the article of Getzler [Ge09] We believe that this is the right generalization ofthe calculus of Lie algebra valued differential forms needed for ordinary principalG-bundles We find a curious 3-form term (see equation (2)) in the Maurer-Cartan equation for differential forms with values in a semi-strict Lie 2-algebrawhich also appears in [He08] In Section 5 we construct the connection 1-formA0 of Baez-Schreiber from the band of the non-abelian gerbe It is not soclear in [BaSc04] on which differential geometric object the construction of A0

is carried out and we believe that expressing it as the usual iterated integralconstruction on the band (which is an ordinary principal G-bundle ) is ofconceptual importance

Section 6 is the heart of our article and explains the mechanism to trans-form the flat connection A0 into a Hochschild cycle for the differential gradedalgebra CHlowast(Ω

lowastΩlowast) It lives therefore in the Hochschild homology of the al-gebra of Hochschild chains Section 7 recalls from [GTZ09] that ldquoHochschildof Hochschildrdquo-homology is the higher Hochschild homology associated to thetorus T2 We proceed with an explicit expression for (some terms arising in)the holonomy cycle in Section 8

In this article to we try to give a more conceptual approch to the holonomy ofgerbes rather than a computational one as it is introduced [GaRe02] and furtherdevelopped in [TWZ11] Another important difference is that our approachincludes non-abelian gerbes as well

The main theorem of the present article is the construction of the homologycycle representing the holonomy It should be thought of as a 0-cochain onmapping surface (torus in this particular case) space

Theorem 1 Consider a non-abelian principal 2-bundle with trivial band on amanifold M with a structure crossed module micro h rarr g such that the Lie algebrah is abelian Then the connection 1-form A0 of Baez-Schreiber gives rise to acycle P (A0) in the higher Hochschild homology HHT

bull (ΩlowastΩlowast) which corresponds

to the holonomy of the gerbe

As stated before we do not consider the condition that h is abelian as arestriction of generality because up to equivalence it may be achieved for anarbitrary crossed module

By construction the cycle P (A0) is not always trivial ie a boundarybecause it represents the holonomy Observe that for the crossed module id h rarr h we recover the result of [AbZe07] The triviality condition on the bandmay be understood as expressing that the construction is local The gluing of thelocally defined connection 1-forms of Baez and Schreiber to a global connection1-form (see [BaSc04]) should permit to glue our Hochschild cycles

Another subject of further research is to understand that the connection1-form A0 does not only lead to a higher Hochschild cycle with respect to the 2-dimensional torus but actually to higher Hochschild cycles with respect to anycompact topological surface In fact we believe that there is a way to recover

HHΣgbull for a connected compact surface Σg of genus g from HHT

bull

5

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

author(s) concept

Bartels principal 2-bundles withcoherent structure group

Baez-Schreiber global connection 1-formfor principal 2-bundles

Schreiber-Waldorf holonomy in terms oftransport functors

Wockel relation to non-abelianCech cohomology

Ginot-Stienon 2-bundles as Hilsum-Skandalisrsquogeneralized morphisms

Martin-Picken connections and holonomy forspecial principal 2-bundles

Chatterjee-Lahiri-Sengupta connections and holonomyusing A-horizontal pathsfor a reference 1-form A

Let us also mention the more recent paper by Nikolaus andWaldorf [NiWa11]where the equivalences between some of the above incarnations of non-abeliangerbes and principal 2-bundles are shown

The goal of our article is to construct a cycle in higher Hochschild homologywhich represents 2-holonomy of a non-abelian gerbe as described above in thesame way as the ordinary holonomy gives rise to a cycle in ordinary Hochschildhomology see [AbZe07] This is done using the connection 1-form of Baez-Schreiber [BaSc04] which we construct here from the band of the non-abeliangerbe

A crucial ingredient in our work is the possibility to arrange that in an ar-bitrary crossed module of Lie algebras micro h rarr g the Lie algebra h is abelianup to equivalence of crossed modules This is shown in Section 1 (see [Wa06])The possibility to have h abelian is used in order to obtain a commutativedifferential graded algebra Ωlowast = Ωlowast(MUh) whose higher Hochschild homol-ogy HHT

bull (ΩlowastΩlowast) associated to the 2-dimensional torus T houses the holonomy

cycle We do not know of any definition of higher Hochschild homology for arbi-trary differential graded algebras therefore we believe the reduction to abelianh to be crucial when working with possibly non-abelian gerbes Section 1 alsoprovides a fundamental result on strict Lie 2-algebras directly inspired from[BaCr04] namely we explicitly show that the two classifications of strict Lie 2-algebras in terms of skeletal models (of the associated semi-strict Lie 2-algebra)and in terms of the associated crossed modules coincide

Section 2 reports on crossed modules of Lie groups These play a minorrole in our study because the main ingredient for the connection data is thethe infinitesimal crossed module ie the Lie algebra crossed module Section 3gives the definition of principal 2-bundles with which we work It is taken fromWockelrsquos article [Wo09] together with restrictions from [BaSc04] In Section 4we discuss in general Linfin-valued differential forms on the manifold M based on

4

the article of Getzler [Ge09] We believe that this is the right generalization ofthe calculus of Lie algebra valued differential forms needed for ordinary principalG-bundles We find a curious 3-form term (see equation (2)) in the Maurer-Cartan equation for differential forms with values in a semi-strict Lie 2-algebrawhich also appears in [He08] In Section 5 we construct the connection 1-formA0 of Baez-Schreiber from the band of the non-abelian gerbe It is not soclear in [BaSc04] on which differential geometric object the construction of A0

is carried out and we believe that expressing it as the usual iterated integralconstruction on the band (which is an ordinary principal G-bundle ) is ofconceptual importance

Section 6 is the heart of our article and explains the mechanism to trans-form the flat connection A0 into a Hochschild cycle for the differential gradedalgebra CHlowast(Ω

lowastΩlowast) It lives therefore in the Hochschild homology of the al-gebra of Hochschild chains Section 7 recalls from [GTZ09] that ldquoHochschildof Hochschildrdquo-homology is the higher Hochschild homology associated to thetorus T2 We proceed with an explicit expression for (some terms arising in)the holonomy cycle in Section 8

In this article to we try to give a more conceptual approch to the holonomy ofgerbes rather than a computational one as it is introduced [GaRe02] and furtherdevelopped in [TWZ11] Another important difference is that our approachincludes non-abelian gerbes as well

The main theorem of the present article is the construction of the homologycycle representing the holonomy It should be thought of as a 0-cochain onmapping surface (torus in this particular case) space

Theorem 1 Consider a non-abelian principal 2-bundle with trivial band on amanifold M with a structure crossed module micro h rarr g such that the Lie algebrah is abelian Then the connection 1-form A0 of Baez-Schreiber gives rise to acycle P (A0) in the higher Hochschild homology HHT

bull (ΩlowastΩlowast) which corresponds

to the holonomy of the gerbe

As stated before we do not consider the condition that h is abelian as arestriction of generality because up to equivalence it may be achieved for anarbitrary crossed module

By construction the cycle P (A0) is not always trivial ie a boundarybecause it represents the holonomy Observe that for the crossed module id h rarr h we recover the result of [AbZe07] The triviality condition on the bandmay be understood as expressing that the construction is local The gluing of thelocally defined connection 1-forms of Baez and Schreiber to a global connection1-form (see [BaSc04]) should permit to glue our Hochschild cycles

Another subject of further research is to understand that the connection1-form A0 does not only lead to a higher Hochschild cycle with respect to the 2-dimensional torus but actually to higher Hochschild cycles with respect to anycompact topological surface In fact we believe that there is a way to recover

HHΣgbull for a connected compact surface Σg of genus g from HHT

bull

5

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

the article of Getzler [Ge09] We believe that this is the right generalization ofthe calculus of Lie algebra valued differential forms needed for ordinary principalG-bundles We find a curious 3-form term (see equation (2)) in the Maurer-Cartan equation for differential forms with values in a semi-strict Lie 2-algebrawhich also appears in [He08] In Section 5 we construct the connection 1-formA0 of Baez-Schreiber from the band of the non-abelian gerbe It is not soclear in [BaSc04] on which differential geometric object the construction of A0

is carried out and we believe that expressing it as the usual iterated integralconstruction on the band (which is an ordinary principal G-bundle ) is ofconceptual importance

Section 6 is the heart of our article and explains the mechanism to trans-form the flat connection A0 into a Hochschild cycle for the differential gradedalgebra CHlowast(Ω

lowastΩlowast) It lives therefore in the Hochschild homology of the al-gebra of Hochschild chains Section 7 recalls from [GTZ09] that ldquoHochschildof Hochschildrdquo-homology is the higher Hochschild homology associated to thetorus T2 We proceed with an explicit expression for (some terms arising in)the holonomy cycle in Section 8

In this article to we try to give a more conceptual approch to the holonomy ofgerbes rather than a computational one as it is introduced [GaRe02] and furtherdevelopped in [TWZ11] Another important difference is that our approachincludes non-abelian gerbes as well

The main theorem of the present article is the construction of the homologycycle representing the holonomy It should be thought of as a 0-cochain onmapping surface (torus in this particular case) space

Theorem 1 Consider a non-abelian principal 2-bundle with trivial band on amanifold M with a structure crossed module micro h rarr g such that the Lie algebrah is abelian Then the connection 1-form A0 of Baez-Schreiber gives rise to acycle P (A0) in the higher Hochschild homology HHT

bull (ΩlowastΩlowast) which corresponds

to the holonomy of the gerbe

As stated before we do not consider the condition that h is abelian as arestriction of generality because up to equivalence it may be achieved for anarbitrary crossed module

By construction the cycle P (A0) is not always trivial ie a boundarybecause it represents the holonomy Observe that for the crossed module id h rarr h we recover the result of [AbZe07] The triviality condition on the bandmay be understood as expressing that the construction is local The gluing of thelocally defined connection 1-forms of Baez and Schreiber to a global connection1-form (see [BaSc04]) should permit to glue our Hochschild cycles

Another subject of further research is to understand that the connection1-form A0 does not only lead to a higher Hochschild cycle with respect to the 2-dimensional torus but actually to higher Hochschild cycles with respect to anycompact topological surface In fact we believe that there is a way to recover

HHΣgbull for a connected compact surface Σg of genus g from HHT

bull

5

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Acknowledgements We thank Gregory Ginot for answering a question abouthigher Hochschild homology We also thank Urs Schreiber and Dimitry Royten-berg for answering a question in the context of Section 1

1 Strict Lie 2-algebras and crossed modules

We gather in this section preliminaries on strict Lie 2-algebras and crossedmodules and their relation to semistrict Lie 2-algebras The main result is thepossibility to replace a crossed module micro h rarr g by an equivalent one havingabelian h This will be important for defining holonomy as a cycle in higherHochschild homology

Lie 2-algebras have been the object of different studies see [BaCr04] forsemi-strict Lie 2-algebras or [Ro07] for (general weak) Lie 2-algebras

11 Strict 2-vector spaces

We fix a field K of characteristic 0 in geometrical situations we will alwaystake K = R A 2-vector space V over K is simply a category object in Vectthe category of vector spaces (cf Def 5 in [BaCr04]) This means that Vconsists of a vector space of arrows Vminus1 a vector space of objects V0 linear

maps Vminus1

s 983587983587t983587983587 V0 called source and target a linear map i V0 rarr Vminus1 called

object inclusion and a linear map

m Vminus1 timesV0Vminus1 rarr Vminus1

which is called the categorical composition These data is supposed to satisfythe usual axioms of a category

An equivalent point of view is to regard a 2-vector space as a 2-term complexof vector spaces d Cminus1 rarr C0 Pay attention to the change in degree withrespect to [BaCr04] We use here a cohomological convention instead of theirhomological convention in order to have the right degrees for the differentialforms with values in crossed modules later on

The equivalence between 2-vector spaces and 2-term complexes is spelt outin Section 3 of [BaCr04] one passes from a category object in Vect (given

by Vminus1

s 983587983587t983587983587 V0 i V0 rarr Vminus1 etc) to a 2-term complex d Cminus1 rarr C0 by

taking Cminus1 = ker(s) d = t|ker(s) and C0 = V0 In the reverse directionto a given 2-term complex d Cminus1 rarr C0 one associates Vminus1 = C0 oplus Cminus1V0 = C0 s(c0 cminus1) = c0 t(c0 cminus1) = c0 + d(cminus1) and i(c0) = (c0 0) The onlysubtle point is here that the categorical composition m is already determined by

Vminus1

s 983587983587t983587983587 V0 and i V0 rarr Vminus1 (see Lemma 6 in [BaCr04]) Namely writing

an arrow cminus1 = f with s(f) = x t(f) = y ie f x 983041rarr y one denotes the

arrow part of f by 983187f = f minus i(s(f)) and for two composable arrows f g isin Vminus1

6

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

the composition m is then defined by

f g = m(f g) = i(x) + 983187f + 983187g

12 Strict Lie 2-algebras and crossed modules

Definition 1 A strict Lie 2-algebra is a category object in the category Lie ofLie algebras over K

This means that it is the data of two Lie algebras g0 the Lie algebra ofobjects and gminus1 the Lie algebra of arrows together with morphisms of Liealgebras s t gminus1 rarr g0 source and target a morphism i g0 rarr gminus1 the objectinclusion and a morphism m gminus1 timesg0 gminus1 rarr gminus1 the composition of arrowssuch that the usual axioms of a category are satisfied

Let us now come to crossed modules of Lie algebras We refer to [Wa06] formore details

Definition 2 A crossed module of Lie algebras is a morphism of Lie algebrasmicro h rarr g together with an action of g on h by derivations such that for allh hprime isin h and all g isin g

(a) micro(g middot h) = [g micro(h)] and

(b) micro(h) middot hprime = [h hprime]

One may associate to a crossed module of Lie algebras a 4-term exact se-quence of Lie algebras

0 rarr V rarr hmicrorarr g rarr g rarr 0

where we used the notation V = ker(micro) and g = coker(micro) It follows from theproperties (a) and (b) of a crossed module that micro(h) is an ideal so g is a Liealgebra and that V is a central ideal of h and a g-module (because the outeraction to be defined below is a genuine action on the center of h)

Recall the definition of the outer action s g rarr out(h) for a crossed moduleof Lie algebras micro h rarr g The Lie algebra

out(h) = der(h)ad(h)

is the Lie algebra of outer derivations of h ie the quotient of the Lie algebraof all derivations der(h) by the ideal ad(h) of inner derivations ie those of theform hprime 983041rarr [h hprime] for some h isin h

To define s choose a linear section ρ g rarr g and compute its default to bea homomorphism of Lie algebras

α(x y) = [ρ(x) ρ(y)]minus ρ([x y])

for x y isin g As the projection onto g is a homomorphism of Lie algebrasα(x y) is in its kernel and there exists therefore an element β(x y) isin h suchthat micro(β(x y)) = α(x y)

7

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

We have for all h isin h

983043ρ(x)ρ(y)minusρ(y)ρ(x)minusρ([x y])

983044middoth = α(x y) middoth = micro(β(x y)) middoth = [β(x y) h]

and in this sense elements of g act on h up to inner derivations We obtain awell defined homomorphism of Lie algebras

s g rarr out(h)

by s(x)(h) = ρ(x) middot hStrict Lie 2-algebras are in one-to-one correspondance with crossed modules

of Lie algebras like in the case of groups cf [Lo82] For the convenience of thereader let us include this here

Theorem 2 Strict Lie 2-algebras are in one-to-one correspondence with crossedmodules of Lie algebras

Proof Given a Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 the corresponding

crossed module is defined by

micro = t|ker(s) h = ker(s) rarr g = g0

The action of g on h is given by

g middot h = [i(g) h]

for g isin g and h isin h (where the bracket is taken in gminus1) This is well definedand an action by derivations Axiom (a) follows from

micro(g middot h) = micro([i(g) h]) = [micro i(g) micro(h)] = [g micro(h)]

Axiom (b) follows from

micro(h) middot hprime = [i micro(h) hprime] = [i t(h) hprime]

by writing i t(h) = h+ r for r isin ker(t) and by using that ker(t) and ker(s) ina Lie 2-algebra commute (shown in Lemma 1 after the proof)

On the other hand given a crossed module of Lie algebras micro h rarr gassociate to it

h⋊ gs 983587983587t983587983587 g i g rarr h⋊ g

by s(h g) = g t(h g) = micro(h) + g i(g) = (0 g) where the semi-direct productLie algebra h ⋊ g is built from the given action of g on h Let us emphasizethat h⋊ g is built from the Lie algebra g and the g-module h the bracket of hdoes not intervene here The composition of arrows is already encoded in theunderlying structure of 2-vector space as remarked in the previous subsection

8

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Lemma 1 [ker(s) ker(t)] = 0 in a Lie 2-algebra

Proof The fact that the composition of arrows is a homomorphism of Liealgebras gives the following ldquomiddle four exchangerdquo (or functoriality) property

[g1 g2] [f1 f2] = [g1 f1 g2 f2]

for composable arrows f1 f2 g1 g2 isin g1 Now suppose that g1 isin ker(s) andf2 isin ker(t) Then denote by f1 and by g2 the identity (with respect to thecomposition) in 0 isin g0 As these are identities we have g1 = g1 f1 andf2 = g2 f2 On the other hand i is a morphism of Lie algebras and sends0 isin g0 to the 0 isin g1 Therefore we may conclude

[g1 f2] = [g1 f1 g2 f2] = [g1 g2] [f1 f2] = 0

Furthermore it is well-known (cf [Wa06]) that (equivalence classes of) crossedmodules of Lie algebras are classified by third cohomology classes H3(g V )

Remark 1It is implicit in the previous proof that starting from a crossed module micro h rarr g

passing to the Lie 2-algebra gminus1

s 983587983587t983587983587 g0 i g0 rarr gminus1 (and thus forgetting

the bracket on h ) one may finally reconstruct the bracket on h This is due tothe fact that it is encoded in the action and the morphism using the property(b) of a crossed module

13 Semi-strict Lie 2-algebras and 2-term Linfin-algebras

An equivalent point of view is to regard a strict Lie 2-algebra as a Lie algebraobject in the category Cat of (small) categories From this second point ofview we have a functorial Lie bracket which is supposed to be antisymmetricand must fulfill the Jacobi identity Weakening the antisymmetry axiom andthe Jacobi identity up to coherent isomorphisms leads then to semi-strict Lie2-algebras (here antisymmetry holds strictly but Jacobi is weakened) hemi-strict Lie 2-algebras (here Jacobi holds strictly but antisymmetry is weakened)or even to (general) Lie 2-algebras (both axioms are weakened) Let us recordthe definition of a semi-strict Lie 2-algebra (see [BaCr04] Def 22)

Definition 3 A semi-strict Lie 2-algebra consists a 2-vector space L togetherwith a skew-symmetric bilinear and functorial bracket [ ] L times L rarr L and acompletely antisymmetric trilinear natural isomorphism

Jxyz [[x y] z] rarr [x [y z]] + [[x z] y]

called the Jacobiator The Jacobiator is required to satisfy the Jacobiator identity(see [BaCr04] Def 22)

9

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Semi-strict Lie 2-algebras together with morphisms of semi-strict Lie 2-algebras (see Def 23 in [BaCr04]) form a strict 2-category (see Prop 25 in[BaCr04]) Strict Lie algebras form a full sub-2-category of this 2-category seeProp 42 in [BaCr04] In order to regard a strict Lie 2-algebra gminus1 rarr g0 as asemi-strict Lie 2-algebra the functorial bracket is constructed for f x 983041rarr y andg a 983041rarr b f g isin gminus1 and x y a b isin g0 by defining its source s([f g]) and its

arrow part 983187[f g] to be s([f g]) = [x a] and 983187[f g] = [x983187g] + [983187f b] (see proof ofThm 36 in [BaCr04]) By construction it is compatible with the compositionie functorial

Remark 2One observes that the functorial bracket on a strict Lie 2-algebra gminus1 rarr g0 isconstructed from the bracket in g0 and the bracket between gminus1 and g0 butdoes not involve the bracket on gminus1 itself

There is a 2-vector space underlying every semi-strict Lie 2-algebras thusone may ask which structure is inherited from a semi-strict Lie 2-algebra by thecorresponding 2-term complex of vector spaces This leads us to 2-term Linfin-algebras see [BaCr04] Thm 36 Our definition here differs from theirs as westick to the cohomological setting and degree +1 differentials see [Ge09] Def41

Definition 4 An Linfin-algebra is a graded vector space L together with a sequencelk(x1 xk) k gt 0 of graded antisymmetric operations of degree 2 minus k suchthat the following identity is satisfied

n983131

k=1

(minus1)k983131

i1ltltik j1ltltjnminusk

i1ikcupj1jnminusk=1n

(minus1)983171ln(lk(xi1 xik) xj1 xjnminusk) = 0

Here the sign (minus1)983171 equals the product of the sign of the shuffle permutationand the Koszul sign We refer the reader to [Stash92] for the definition of Linfin-morphism

We will be mainly concerned with 2-term Linfin-algebras These are Linfin-algebras L such that the graded vector space L consists only of two componentsL0 and Lminus1 An Linfin-algebra L = L0 oplusLminus1 has at most l1 l2 and l3 as its non-trivial ldquobracketsrdquo l1 is a differential (ie here just a linear map L0 rarr Lminus1)l2 is a bracket with components [ ] L0 otimes L0 rarr L0 and [ ] Lminus1 otimes L0 rarr Lminus1[ ] L0 timesLminus1 rarr Lminus1 and l3 is some kind of 3-cocycle l3 L0 otimesL0 otimesL0 rarr Lminus1More precisely in case l1 = 0 L0 is a Lie algebra Lminus1 is an L0-module and l3 isthen an actual 3-cocycle This kind of 2-term Linfin-algebra is called skeletal seeSection 6 in [BaCr04] and our next subsection The complete axioms satisfiedby l1 l2 and l3 in a 2-term Linfin-algebra are listed in Lemma 33 of [BaCr04]

As said before the passage from a 2-vector space to its associated 2-termcomplex induces a passage from semi-strict Lie 2-algebras to 2-term Linfin-algebraswhich turns out to be an equivalence of 2 categories (see [BaCr04] Thm 36)

10

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Theorem 3 The 2-categories of semi-strict Lie 2-algebras and of 2-term Linfin-algebras are equivalent

Remark 3In particular restricting to the sub-2-category of strict Lie 2 algebras there isan equivalence between crossed modules of Lie algebras and 2-term Linfin-algebraswith trivial l3 In other words there is an equivalence between crossed modulesand differential graded Lie algebras Lminus1 oplus L0

14 Classification of semi-strict Lie 2-algebras

Baez and Crans show in [BaCr04] that every semi-strict Lie 2-algebra is equiva-lent to a skeletal Lie 2-algebra (ie one where the differential d of the underlyingcomplex of vector spaces is zero) Then they go on by showing that skeletal Lie2-algebras are classified by triples consisting of an honest Lie algebra g a g-module V and a class [γ] isin H3(g V ) This is achieved using the homotopyequivalence of the underlying complex of vector spaces with its cohomology Intotal they get in this way a classification up to equivalence of semi-strict Lie2-algebras in terms of triples (g V [γ])

On the other hand strict Lie 2-algebras are in one-to-one correspondancewith crossed modules of Lie algebras as we have seen in a previous subsectionIn conclusion there are two ways to classify strict Lie 2-algebras By the asso-ciated crossed module or regarding them as special semi-strict Lie 2-algebrasby Baez-Crans classification Let us show here that these two classifications arecompatible ie that they lead to the same triple (g V [γ])

For this let us denote by sLie2 the 2-category of strict Lie 2-algebras byssLie2 the 2-category of semi-strict Lie 2-algebras by sssLie2 the 2-category ofskeletal semi-strict Lie 2-algebras by triples the (trivial) 2-category of triplesof the above form (g V [γ]) and by crmod the 2-category of crossed modules ofLie algebras

Theorem 4 The following diagram is commutative

sLie2

inclusion 983578983578

α

983541983541

ssLie2

skeletal model

983555983555crmod

β 983577983577

sssLie2

γ

983660983660983225983225983225983225983225983225

983225983225983225983225983225

triples

The 2-functors α and γ are bijections while the 2-functor β induces a bijec-tion when passing to equivalence classes

11

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Proof Let us first describe the arrows The arrow α sLie2 rarr crmod hasbeen investigated in Theorem 2 The arrow β crmod rarr triplets sends acrossed module micro h rarr g to the triple

(coker(micro) = g ker(micro) = V [γ])

where the cohomology class [γ] isin H3(g V ) is defined choosing sections ndash theprocedure is described in detail in [Wa06] The arrow ssLie2 rarr sssLie2 is thechoice of a skeletal model for a given semi-strict Lie 2-algebra ndash it is given bythe homotopy equivalence of the underlying 2-term complex with its cohomologydisplayed in the extremal lines of the following diagram

Cminus1d 983587983587 C0

ker(d)983603983571

983619983619

0 983587983587 C0

983555983555983555983555ker(d)

0 983587983587 C0 im(d)

More precisely choosing supplementary subspaces one can define a map ofcomplexes of vector spaces from 0 ker(d) rarr C0 im(d) to d Cminus1 rarr C0 andgive the structure of an Linfin-algebra to the latter in such a way that it becomesa morphism (with a new 3-cocycle lprime3 that differs from the given 3-cocycle l3 atmost by a coboundary) see below

The arrow γ sssLie2 rarr triples sends a skeletal 2-Lie algebra to thetriple defined by the cohomology class of l3 (cf [BaCr04])

Now let us show that the diagram commutes For this let d Cminus1 rarr C0

with some bracket [ ] and l3 = 0 be a 2-term Linfin-algebra corresponding toseeing a strict Lie 2-algebra as a semi-strict Lie 2-algebra and build its skeletalmodel The model comes together with a morphism of semi-strict Lie 2-algebras(φ2φminus1φ0) given by

Cminus1d 983587983587 C0

ker(d)983603983571

φminus1

983619983619

0 983587983587 C0 im(d)

φ0

983619983619

Here φ0 = σ is a linear section of the quotient map The structure of a semi-strict Lie 2-algebra is transfered to the lower line in order to make (φ2φminus1φ0)a morphism of semi-strict Lie 2-algebras In order to compute now the l3 term ofthe lower semi-strict Lie 2-algebra denoted lprime3 one first finds that (first equationin Definition 34 of [BaCr04]) φ2 C0 im(d)times C0 im(d) rarr Cminus1 is such that

dφ2(x y) = σ[x y]minus [σ(x)σ(y)]

12

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

the default of the section σ to be a homomorphism of Lie algebras Then lprime3is related to φ2 by the second formula in Definition 34 of [BaCr04] This giveshere

lprime3(x y z) = (dCEφ2)(x y z)

for x y z isin C0 im(d) dCE is the formal Chevalley-Eilenberg differential of thecochain φ2 C0 im(d)times C0 im(d) rarr Cminus1 with values in Cminus1 as if Cminus1 was aC0 im(d)-module (which is usually not the case) This is exactly the expressionof the cocycle γ associated to the crossed module of Lie algebras d Cminus1 rarr C0

obtained using the section σ see [Wa06]

Corollary 1 Every semi-strict Lie 2-algebra is equivalent (as an object of the2-category ssLie2) to a strict Lie 2-algebra

This corollary is already known because of abstract reasons Here we haveproved a result somewhat more refined The procedure to strictify a semi-strictLie 2-algebra is rather easy to perform First one has to pass to cohomology byhomotopy equivalence (ie the arrows ldquoskeletal modelrdquo and γ in the diagram ofTheorem 4) and then one has to construct the crossed module correspondingto a given cohomology class This can be done in several ways using free Liealgebras [LoKa82] using injective modules [Wa06] (as described in the nextsubsection) etc and one may adapt the construction method to the problem athand

15 The construction of an abelian representative

We will show in this section that to a given class [γ] isin H3(g V ) there exists acrossed module of Lie algebras micro h rarr g with class [γ] (and ker(micro) = V andcoker(micro) = g) such that h is abelian This will be important for the treatmentin higher Hochschild homology of the holonomy of a gerbe

Theorem 5 For any [γ] isin H3(g V ) there exists a crossed module of Lie alge-bras micro h rarr g with associated class [γ] such that ker(micro) = V coker(micro) = g andh is abelian

Proof This is Theorem 3 in [Wa06] Let us sketch its proof here The cat-egory of g-modules has enough injectives therefore V may be embedded in aninjective g-module I We obtain a short exact sequence of g-modules

0 rarr Virarr I

πrarr Q rarr 0

where Q = IV is the quotient I injective implies Hp(g I) = 0 for all p gt 0Therefore the short exact sequence of coefficients induces a connective homo-morphism

part H2(g Q) rarr H3(g V )

13

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

which is an isomorphism To [γ] corresponds thus a class [α] isin H2(g Q) withpart[α] = [γ] A representative α isin Z2(g Q) gives rise to an abelian extension

0 rarr Q rarr Qtimesα g rarr g rarr 0

Now one easily verifies (see the proof of Theorem 3 in [Wa06]) that the splicingtogether of the short exact coefficient sequence and the abelian extension givesrise to a crossed module

0 rarr V rarr I rarr Qtimesα g rarr g rarr 0

More precisely the crossed module is micro I rarr Qtimesα g given by micro(x) = (π(x) 0)the action of g = Qtimesα g on h = I is induced by the action of g on I and theLie bracket is trivial on I ie I is abelian

One also easily verifies (see the proof of Theorem 3 in [Wa06]) that the asso-ciated cohomology class for such a crossed module (which is the Yoneda productof a short exact coefficient sequence and an abelian extension) is part[α] the im-age under the connective homomorphism (induced by the short exact coefficentsequence) of the class defining the abelian extension Therefore the associatedclass is here part[α] = [γ] as required

We thus obtain the following refinement of Corollary 1

Corollary 2 Every semi-strict Lie 2-algebra is equivalent to a strict Lie 2-algebra corresponding to a crossed module micro h rarr g with abelian h such that his a g = gmicro(h)-module and such that the outer action is a genuine action

Proof This follows from Corollary 1 together with Theorem 5 The fact thath is a g = gmicro(h)-module and that the outer action is a genuine action areequivalent They are true either by inspection of the representative constructedin the proof of Theorem 5 or by the following argument

The outer action s is an action only up to inner derivations But these aretrivial in case h is abelian

micro(h) middot hprime = [h hprime] = 0

for all h hprime isin h by property (b) of a crossed module

Remark 4An analoguous statement is true on the level of (abstract) groups and even topo-logical groups [WaWo11] Unfortunately we ignore whether such a statement istrue in the category of Lie groups ie given a locally smooth group 3-cocycle γon G with values in a smooth G-module V is there a smooth (not necessarilysplit) crossed module of Lie groups micro H rarr G with H abelian and cohomologyclass [γ] From the point of view of Lie algebras there are two steps involvedhaving solved the problem on the level of Lie algebras (as above) one has tointegrate the 2-cocycle α This is well-understood thanks to work of Neeb The

14

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

(possible) obstructions lie in π1(G) and π2(G) and vanish thus for simply con-nected finite dimensional Lie groups G The second step is to integrate theinvolved g-module I to a G-module As I is necessarily infinite dimensionalthis is the hard part of the problem

Note however that for many interesting crossed modules it is not necessaryto perform the construction using a genuine injective module I The only thingwe need about I is that V sub I and H3(g I) = 0 because then the connectinghomomorphism is surjective In concrete situations there are often much easiermodules which have this property Note furthermore that for many interestingclasses of crossed modules of Lie algebras micro m rarr n m is already abelianThis is the case for id m rarr m with m abelian or for 0 V rarr g where V is ag-module

2 Crossed modules of Lie groups

In this section we introduce the strict Lie 2-groups which will be the typicalfiber of our principal 2-bundles While the notion of a crossed module of groupsis well-understood and purely algebraic the notion of a crossed module of Liegroups involves subtle smoothness requirements

We will heavily draw on [Ne07] and adopt Neebrsquos point of view namelywe regard a crossed module of Lie groups as a central extension N rarr N of anormal split Lie subgroup N in a Lie group G for which the conjugation actionof G on N lifts to a smooth action on N This point of view is linked to the oneregarding a crossed module as a homomorphism micro H rarr G by taking H = Nand im(micro) = N

Definition 5 A morphism of Lie groups micro H rarr G together with a ho-momorphism S G rarr Aut(H) defining a smooth action S G times H rarr H(g h) 983041rarr g middot h = S(g)(h) of G on H is called a (split) crossed module of Liegroups if the following conditions are satisfied

1 micro S(g) = conjmicro(g) micro for all g isin G

2 S micro H rarr Aut(H) is the conjugation action

3 ker(micro) is a split Lie subgroup of H and im(micro) is a split Lie subgroup of Gfor which micro induces an isomorphism Hker(micro) rarr im(micro)

Recall that in a split crossed module of Lie groups micro H rarr G the quotientLie group G = Gmicro(H) acts smoothly (up to inner automorphisms) on HThis outer action S of G on H is a homomorphism S G rarr Out(H) whichis constructed like in the case of Lie algebras The smoothness of S followsdirectly from the splitting assumptions Here Out(H) denotes the group ofouter automorphisms of H defined by

Out(H) = Aut(H)Inn(H)

15

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

where Inn(H) sub Aut(H) is the normal subgroup of automorphisms of the formhprime 983041rarr hhprimehminus1 for some h isin H

It is shown in [Ne07] that one may associate to a (split) crossed module ofLie groups a locally smooth 3-cocycle γ (whose class is the obstruction againstthe realization of the outer action in terms of an extension)

It is clear that a (split) crossed module of Lie groups induces a crossedmodule of the corresponding Lie algebras

Definition 6 Two crossed modules micro M rarr N (with action η) and microprime M prime rarrN prime (with action ηprime) such that ker(micro) = ker(microprime) = V and coker(micro) = coker(microprime) =G are called elementary equivalent if there are group homomorphisms ϕ M rarrM prime and ψ N rarr N prime which are compatible with the actions ie

ϕ(η(n)(m)) = ηprime(ψ(n))(ϕ(m))

for all n isin N and all m isin M and such that the following diagram is commuta-tive

0 983587983587 V

idV

983555983555

i 983587983587 M

ϕ

983555983555

micro 983587983587 N

ψ

983555983555

π 983587983587 G

idG

983555983555

983587983587 1

0 983587983587 Viprime 983587983587 M prime microprime

983587983587 N prime πprime983587983587 G 983587983587 1

We call equivalence of crossed modules the equivalence relation generated byelementary equivalence One easily sees that two crossed modules are equivalentin case there exists a zig-zag of elementary equivalences going from one to theother (where the arrows do not necessarily all go into the same direction)

In the context of split crossed modules of Lie groups all morphisms aresupposed to be morphisms of Lie groups ie smooth and to respect the sections

Remark 5It is elementary to show that a morphism of crossed modules of finite dimen-sional Lie algebras micro m rarr n integrates to a morphism of crossed modules ofLie groups M rarr N for the 1-connected Lie groups M N corresponding to mresp n This implies in particular that under this hypothesis equivalences ofcrossed modules of Lie algebras integrate to equivalences of crossed modules ofLie groups

3 Principal 2-bundles and gerbes

In this section we will start introducing the basic geometric objects of ourstudy namely principal 2-bundles and gerbes We choose to work here with astrict Lie 2-group G ie a split crossed module of Lie groups and its associatedcrossed module of Lie algebras micro h rarr g and to consider principal 2-bundlesand gerbes which are defined by non-abelian cocycles (or transition functions)The principal object which we will use later on is the band of a gerbe

16

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

31 Definition

In order to keep notations and abstraction to a reasonable minimum we willconsider geometric objects like bundles gerbes etc only over an honest (finitedimensional) base manifold M instead of considering a ringed topos or a stack

Let micro H rarr G be a (split) crossed module of Lie groups Let our base spaceM be an honest (ordinary) manifold and let U = Ui be a good open coverof M The following definition dates back at least to [BreMe] in the presentform we took it from [BaSc04] p 29 (see also the corresponding presentationin [Wo09])

Definition 7 A non-abelian cocycle (gij hijk) is the data of (smooth) transitionfunctions

gij Ui cap Uj rarr G

andhijk Ui cap Uj cap Uk rarr H

which satisfy the non-abelian cocycle identities

micro(hijk(x))gij(x)gjk(x) = gik(x)

for all x isin Uijk = Ui cap Uj cap Uk and

hikl(x)hijk(x) = hijl(x)(gij(x) middot hjkl(x))

for all x isin Uijkl = Ui cap Uj cap Uk cap Ul

The Cech cochains gij and hijk are (by definition) ordered in the indicesone may then extend to antisymmetric indices One may furthermore completethe set of indices to all pairs resp triples by imposing the functions to be equalto 1G resp 1H on repeated indices

We go on by defining equivalence of non-abelian cocycles with values in thesame crossed module of Lie groups micro H rarr G

Definition 8 Two non-abelian cocycles (gij hijk) and (gprimeij hprimeijk) on the same

cover are said to be equivalent if there exist (smooth) functions γi Ui rarr G andηij Uij rarr H such that

γi(x)gprimeij(x) = micro(ηij(x))gij(x)γj(x)

for all x isin Uij and

ηik(x)hijk(x) = (γi(x) middot hprimeijk(x))ηij(x)(gij(x) middot ηjk(x))

for all x isin Uijk

In general one should define equivalence for cocycles corresponding to differentcovers Passing to a common refinement one easily adapts the above definitionto this framework (this is spelt out in [Wo09]) Furthermore one also definesequivalence for cocycles corresponding to different crossed modules see eg[NiWa11]

17

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Definition 9 A principal 2-bundle also called (non-abelian) gerbe and denotedG is the data of an equivalence class of non-abelian cocycles

By abuse of language we will also call a representative (gij hijk) a principal2-bundle or a (non-abelian) gerbe

Lemma 2 If the (split) crossed module of Lie groups micro H rarr G is replacedby an equivalent crossed module microprime H prime rarr Gprime then the corresponding principal2-bundles are equivalent

Proof This follows from Theorem 623 in [NiWa11]

This lemma is important for us because in case we replace (infinitesimal)crossed modules by equivalent ones we want to obtain an equivalent principal2-bundle

Recall that for a split crossed module of Lie groups micro H rarr G the imagemicro(H) is a normal Lie subgroup of G and the quotient group G = Gmicro(H) istherefore a Lie group

Lemma 3 Let G be a gerbe defined by the cocycle (gij hijk)Then one may associate to G an ordinary principal G-bundle B on M which

has as its transition functions the composition of the gij and the canonical pro-jection G rarr Gmicro(H) = G

Proof This is clear Indeed passing to the quotient G rarr Gmicro(H) theidentity

micro(hijk(x))gij(x)gjk(x) = gik(x)

becomes the cocycle identity

gij(x)gjk(x) = gik(x)

for a principal Gmicro(H)-bundle on M defined by the transition functions

gij Uij rarr Gmicro(H)

obtained from composing gij Uij rarr G with the projection G rarr Gmicro(H)

Definition 10 The principal G-bundle B on M associated to the gerbe G definedby the cocycle (gij hijk) is called the band of G

32 Connection data

Let as before M be a manifold and let U = Ui be a good open cover ofM Let G be a gerbe defined by the cocycle (gij hijk) We associate to G nowconnection data like in [BaSc04] Sect 214

18

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Definition 11 Connection data for the non-abelian cocycle (gij hijk) is thedata of connection 1-forms Ai isin Ω1(Ui g) and of curving 2-forms Bi isin Ω2(Ui h)together with connection transformation 1-forms aij isin Ω(Uij h) and curvingtransformation 2-forms dij isin Ω2(Uij h) such that the following laws hold

(a) transition law for connection 1-forms on Uij

Ai + micro(aij) = gijAjgminus1ij + gijdg

minus1ij

(b) transition law for the curving 2-forms on Uij

Bi = gij middotBj + daij

(c) transition law for the curving transformation 2-forms on Uijk

dij + gij middot djk = hijkdikhminus1ijk + hijk(micro(Bi) + FAi)h

minus1ijk

(d) coherence law for the transformers of connection 1-forms on Uijk

0 = aij + gij middot ajk minus hijkaikhminus1ijk minus hijkdh

minus1ijk minus hijk(Ai middot hminus1

ijk)

In accordance with [BaSc04] equation (273) on p 59 we will choose dij = 0in the following The transition law (c) for the curving transformation 2-formsreads then simply

0 = micro(Bi) + FAi

which is the equation of vanishing fake curvature In the following we willalways suppose that the fake curvature vanishes (cf Section 4)

Definition 12 Let G be a gerbe defined by the cocycle (gij hijk) with connectiondata (Ai Bi aij) Then the curvature 3-form Hi isin Ω3(Ui h) is defined by

Hi = dAiBi

ie it is the covariant derivative of the curving 2-form Bi isin Ω2(Ui h) withrespect to the connection 1-form Ai isin Ω1(Ui h)

Its transformation law on Uij is

Hi = gij middotHj

(because in our setting fake curvature and curving transformation 2-forms van-ish)

Observe that only the crossed module of Lie algebras micro h rarr g plays arole as values of the differential forms Ai and Bi According to Section 1 itconstitutes no restriction of generality (up to equivalence) to consider h abelianIf all components in the crossed module are finite dimensional this equivalenceinduces even an equivalence between the corresponding crossed modules of 1-connected Lie groups In our main application (construction of the holonomy

19

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

higher Hochschild cycle) we will suppose h to be abelian Many steps on theway are true for arbitrary h The property of being abelian simplifies the abovecoherence law (d) for the transformers of connection 1-forms on Uijk for whichwe thus obtain in the abelian setting

0 = aij + gij middot ajk minus aik minus hijkdhminus1ijk minus hijk(Ai middot hminus1

ijk)

We note in passing

Lemma 4 The connection 1-forms induce an ordinary connection on the bandB of the gerbe G

Proof This follows at once from equation (a) in the definition of connectiondata

On the other hand we will always be in a local setting therefore in thefollowing we will drop the indices i j k which refer to the open set we areon

4 Linfin-valued differential forms

In this section we will associate to each principal 2-bundle an Linfin-algebra ofLinfin-valued differential forms This Linfin-algebra replaces the differential gradedLie algebra of Lie algebra valued forms which plays a role for ordinary principalG-bundles Here the Linfin-algebra of values (of the differential forms) will bethe 2-term Linfin-algebra associated with the strict structure Lie 2-algebra of theprincipal 2-bundle We follow closely [Ge09] section 4

Given an Linfin-algebra ginfin and a manifold M the tensor product Ωlowast(M)otimesginfinof ginfin-valued smooth differential forms on M is an Linfin-algebra by prolongatingthe Linfin-operations of ginfin point by point to differential forms The only point tonotice is that the de Rham differential ddeRham gives a contribution to the firstbracket l1 ginfin rarr ginfin which is also a differential of degree 1

We will apply this scheme to the 2-term Linfin-algebras arising from a semi-strict Lie 2-algebra ginfin = (gminus1 g0) The only (possibly) non-zero operationsare the differential l1 the bracket [ ] = l2 and the 3-cocycle l3 Our choiceof degrees is that an element αk isin Ωlowast(M) otimes ginfin is of degree k in case αk isin983119

ige0 Ωi(M) otimes gkminusi An element of degree 1 is thus a sum α1 = α1 + α2 with

α1 isin Ω1(M)otimes g0 and α2 isin Ω2(M)otimes gminus1Recall the following definitions (cf Def 42 in [Ge09])

Definition 13 The MaurerndashCartan set MC(ginfin) of a nilpotent Linfin-algebra ginfinis the set of α isin g1 satisfying the MaurerndashCartan equation F(α) = 0 Moreexplicitly this means

F(α) = l1α+

infin983131

k=2

1

klk(α α) = 0

20

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

The MaurerndashCartan equations for the degree 1 elements of the Linfin-algebraΩlowast(M)otimes ginfin (see [Ge09] Def 43) with ginfin = (gminus1 g0) read therefore

ddeRhamα1 +1

2[α1α1] + l1α2 = 0 (1)

andddeRhamα2 + [α1α2] + l3(α1α1α1) = 0 (2)

Equation (1) is an equation of 2-forms in the gerbe literature it is known asthe equation of the vanishing of the fake curvature Equation (2) is an equationof 3-forms and seems to be (kind of) new in this context (It appears in [He08]Section 8) The special case l3 = 0 corresponds to example 6513 in [SSS09]when one interpretes ddeRhamα2 + [α1α2] as the covariant derivative dα1

α2When applied to connection data of a non-abelian gerbe (see Section 32) thevanishing of the covariant derivative means that the 3-curvature (cf Definition12) of the gerbe vanishes This is sometimes expressed as being a flat gerbe

Let us record the special case of a strict Lie 2-algebra ginfin given by a crossedmodule micro h rarr g for later use

Lemma 5 A degree 1 element of the Linfin-algebra Ωlowast(M)otimes ginfin is a pair (AB)with A isin Ω1(M)otimes g and B isin Ω2(M)otimes h

The element (AB) satisfies the Maurer-Cartan equation if and only if

ddeRhamA+1

2[AA] + microB = 0 and ddeRhamB + [AB] = 0

Elements of degree 0 in Ωlowast(M) otimes g are sums α0 = β0 + β1 with β0 isinΩ0(M) otimes g0 and β1 isin Ω1(M) otimes gminus1 These act by gauge transformations onelements of the Maurer-Cartan set Namely β0 has to be exponentiated to anelement B0 isin Ω0(MG0) (where G0 is the connected 1-connected Lie groupcorresponding to g0) and leads then to gauge transformations of the first kindin the sense of [BaSc04] Elements β1 lead directly to gauge transformations ofthe second kind in the sense of [BaSc04] The fact that they do not have to beexponentiated corresponds to the fact that there is no bracket on the gminus1-part ofthe Linfin-algebra These gauge transformations will not play a role in the presentpaper but will become a central subject when gluing the local expressions ofthe connection 1-form of Baez-Schreiber to a global connection

Definition 14 Let g be a nilpotent Linfin-algebra The MaurerndashCartan varietyMC(g) is the quotient of the MaurerndashCartan set MC(g) by the exponentiatedaction of the infinitesimal automorphisms g0 of MC(g)

We do not assert that the quotientMC(g) is indeed a variety It is consideredhere as a set

21

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

5 Path space and the connection 1-form associ-ated to a principal 2-bundle

In this section we explain how the connective structure on a gerbe gives rise toa connection on path space

51 Path space as a Frechet manifold

We first recall some basic facts about path spaces which allow us to employthe basic notions of differential geometry in particular differential forms andconnections to path spaces For a manifold M let PM = Cinfin([0 1]M) bethe space of paths in M Baez and Schreiber [BaSc04] fix in their definitionthe starting point and the end point of the paths ie for two points s and tin M Pt

sM denotes the space of paths from s (ldquosourcerdquo) to t (ldquotargetrdquo) Thespace PM can made into a Frechet manifold modeled on the Frechet spaceCinfin([0 1]Rn) in case n is the dimension of M Similar constructions exhibitthe loop space

LM = γ [0 1]Cinfin

rarr M | γ(0) = γ(1) γ(k)(0) = γ(k)(1) = 0 forallk ge 1

as a Frechet manifold see for example [Ne04] for a detailed account on theFrechet manifold structure on (this version of) LM in case M is a Lie groupThe generalization to arbirary M is quite standard Let us emphasize that thisversion of LM comes to mind naturally when writing the circle S1 as [0 1] simin Cinfin(S1M) The fact that one demands γ(k)(0) = γ(k)(1) = 0 and not onlyγ(k)(0) = γ(k)(1) for all k ge 1 is sometimes expressed by saying that the loopshave a ldquositting instantrdquo

For differentiable Frechet manifolds one can introduce differential formsde Rham differential and prove a De Rham theorem for smoothly paracompactFrechet manifolds The only thing beyond the necessary definitions that we needfrom Frechet differential geometry is an expression of the de Rham differentialon LM an expression due to Chen [Ch73] which will play its role in the proofof Proposition 3

52 The connection 1-form of Baez-Schreiber

Let micro H rarr G be a split crossed module of Lie groups Denote by S the outeraction of G on H ie the homomorphism S G rarr Out(H)

Composing the transition functions gij Uij rarr G with the homomorphismS G rarr Out(H) we obtain the transition functions of an Out(H)-principalbundle denoted BS This is then an ordinary principal bundle and we mayapply ordinary holonomy theory to the principal bundle BS

As we are only interested in these constructions and these constructions arepurely on Lie algebra level we will neglect now the crossed module of Lie groupsand focus on the crossed module of Lie algebras In doing so we may assume (upto equivalence without loss of generality) that in the crossed module micro h rarr g

22

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

h is abelian and that the outer action s g rarr out(h) (associated to micro h rarr glike in Section 1) is a genuine action (see Corollary 2) Note that we thus haveout(h) = der(h) = End(h) again because h is abelian

In the following we will suppose that the principal bundle BS is trivial (orin other words we will do a local construction) A connection 1-form on BS

is then simply a differential form AS isin Ω1(MEnd(h)) and given a 1-formA isin Ω1(M g) one obtains such a form AS by AS = s A We will supposethat BS posseses a flat connection nabla which will be our reference point in theaffine space of connections

Actually in case the 1-form A isin Ω1(M g) (and not in Ω1(M g) ) there isno problem to define the action of A on B We do not need h to be abelian here(in case we do not want to use the band for example)

Consider now the loop space LM of M Let us proceed with the Wilsonloop or iterated integral construction of Section 6 of [AbZe07] For every n ge 0consider the n-simplex

n = (t0 t1 tn tn+1) | 0 = t0 le t1 le le tn le tn+1 = 1

Define the evaluation maps ev and evni as follows

ev n times LM rarr M

ev(t0 t1 tn tn+1 γ) = γ(0) = γ(1)

evni n times LM rarr M

evni(t0 t1 tn tn+1 γ) = γ(ti)

Denote by adBS the adjoint bundle associated to the principal bundle BS

using the adjoint action of Out(H) on out(h) = der(h) = End(h) Let Ti evlowastni(adBS) rarr evlowast(adBS) denote the map between pullbacks of adjoint bun-dles to n times LM defined at a point (0 = t0 t1 tn tn+1 = 1 γ) by theparallel transport along and in the direction of γ from γ(ti) to γ(tn+1) = γ(1)in the bundle adBS with respect to the flat connection nabla

Denote by BUS the associated bundle to BS with typical fiber the universal

enveloping algebra U End(h)For αi isin Ωlowast(M adBS) 1 le i le n define V n

α1αnisin Ωlowast(LM evlowastBU

S ) by

V 0α1αn

= 1

V nα1αn

=983125n T1ev

lowastn1α1 and and Tnev

lowastnnαi for n ge 1

and set

Vα =

infin983131

n=0

V nαα

It is noteworthy that this infinite sum is convergent This is shown in [AbZe07] inAppendix B Observe that for 1-forms α1 αn the loop space form V n

α1αn

has degree 0 for all nFurthermore define for B isin Ω2(M h) and σ isin [0 1] the 1-form Blowast(σ) isin

Ω1(LM h) byBlowast(σ) = iKEVlowast

σB

23

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

for the evaluation map EVσ LM rarr M EVσ(γ) = γ(σ) and the vector fieldK on LM which is the infinitesimal generator of the S1-action on LM by rigidrotations

Now fix an element (AB) of the Maurer-Cartan set with respect to someLie algebra crossed module micro h rarr g Evaluating elements of U End(h) on hwe obtain a connection 1-form A0 on LM with values in h given by

A0 =

983133 1

0

VA(Blowast(σ))dσ

(Indeed as A is a 1-form the loop space form VA isin Ω0(LM evlowastBUS ) is of

degree 0 ie a evlowastBUS -value function and VA(B

lowast(σ)) is of degree 1 and remainsof degree 1 after integration with respect to σ)

This gives the formula for the connection 1-form of Baez and Schreiber onp 43 of [BaSc04]

Proposition 1 The constructed connection 1-form A0 on LM with values in hcoincides with the path space 1-form A(AB) =

983115A(B) of Def 223 in [BaSc04]

Proof This follows from a step-by-step comparison

6 The holonomy cycle associated to a principal2-bundle

A central construction of [ATZ10] associates to elementsA in the Maurer-Cartanspace a holonomy class [P (A)] in HHlowast(Ω

lowastΩlowast) This is done using the followingproposition (cf loc cit Section 4)

Proposition 2 Suppose given a differential graded associative algebra (Ωlowast d)and an element A isin Ωodd The following are equivalent

(a) A is a Maurer-Cartan element ie dA+A middot A = 0

(b) the chainP (A) = 1otimes 1 + 1otimesA+ 1otimesAotimesA+

in the Hochschild complex CHlowast(ΩlowastΩlowast) is a cycle

Proof We adopt the sign convention of Tsygan [CST04 p78] for differentialgraded Hochschild homology In this convention the total differential dHoch isthe sum of the internal differential

d(a0 otimes otimes ap) =

p983131

i=0

(minus1)1+983123

klti(|ak|+1)a0 otimes otimes dai otimes otimes ap

24

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

and of the (appropriately signed) Hochschild differential

b(a0 otimes otimes ap) =

pminus1983131

k=0

(minus1)1+983123k

i=0(|ai|+1)a0 otimes otimes akak+1 otimes otimes ap +

+(minus1)|ap|+(|ap|+1)983123pminus1

i=0 (|ai|+1)apa0 otimes otimes apminus1

Note that there is no (additional relative) sign between d and b ie the totaldifferential is dHoch = d+ b (and satisfies d2Hoch = 0)

We have to compute dHoch(P (A)) Let us only write down the terms con-tributing to one fixed tensor degree p + 1 These are d(1 otimes Aotimes otimesA983167 983166983165 983168

p times

) and

b(1otimesAotimes otimesA983167 983166983165 983168p+1 times

) We obtain for the sum of these two terms

p983131

i=0

(minus1)1+983123

klti(|A|+1)1otimesA otimes dA983167983166983165983168i th place

otimes otimesA +

+

p983131

i=0

(minus1)1+983123i

k=0(|A|+1)1otimesA otimes A middot A983167 983166983165 983168i th place

otimes otimesA +

+(minus1)|A|+(p+1)(|A|+1)2Aotimes otimesA

Now in this sum there are exactly two terms of the form Aotimes otimesA In casethe degree of A ie |A| is odd these two terms add up to zero On the otherhand the other terms of the sum add to terms of the form

983131plusmn1otimesA otimes (dA+A middot A)otimes otimesA

Therefore the Maurer-Cartan equation implies that P (A) is a cycleIn the reverse direction if P (A) is a cycle then the terms of tensor de-

gree two gives exactly the Maurer-Cartan equation Thus the cycle property isequivalent to the Maurer-Cartan equation

The degrees are taken such that all terms in P (A) are of degree 0 in caseA is of degree 1 ie the degrees of Ωlowast are shifted by one This is the correctdegree when taking Hochschild homology as a model for loop space cohomology

We will apply this proposition to the connection 1-form A0 on LM The1-form A0 is an element of Ω1(LMUh) The condition that A0 is a Maurer-Cartan element is then that the curvature of A0 vanishes This curvature hasbeen computed in [BaSc04] p 43 to be given by the following formula (needlessto say no assumption is made on h for this computation)

25

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Proposition 3 The curvature of the 1-form A0 is equal to

FA0 = minus983116

A

(dAB)minus983116

A

(dα(Ta)(B) (FA + micro(B))a)

=

983133 1

0

VA((dAB)lowast(σ))dσ minus983133 1

0

VA((FA + micro(B))lowast(σ))dσ

where dAB is the covariant derivative of B with respect to A and FA + micro(B) isthe fake curvature of the couple (AB)

Proof A detailed proof is given in [BaSc04] Prop 27 and Cor 22 p 42-43Here we will only sketch the main steps of the proof

First compute ddeRhamA0 for the de Rham differential ddeRham As explainedin loc cit Prop 24 p 35 the action of the de Rham differential on a Chenform

983115A(ω1 ωn) is given by two terms namely

983131

k

plusmn983116

A

(ω1 ddeRhamωk ωn)

and 983131

k

plusmn983116

A

(ω1 ωkminus1 and ωk ωn)

In our case we get thus four terms according to whether the B is involved ornot The terms which do not involve B give a term involving the curvatureFA = dA + A middot A The terms involving B give a term involving the covariantderivative dAB = ddeRhamB +A middotB of B with respect to A

Now the computation of the curvature of A0 adds to the de Rham derivativeddeRhamA0 a term A0 middot A0 This term is easily seen to be the term involvingmicro(B)

Remark 6If f(AB) is a a Maurer-Cartan element (in the sense of Section 4) then (byLemma 5 and Proposition 3) dAB and the fake curvature vanish therefore A0

is a flat connection and P (A0) is a Hochschild cycle (by Proposition 2)

Definition 15 The Hochschild cycle P (A0) is the holonomy cycle associated tothe given principal 2-bundle with connection

Let us abbreviate Ωlowast(MUh) to Ωlowast Our main point is now that the assump-tion that h is abelian implies that Ωlowast (and for the same reason also Ωlowast(LMUh))is a commutative differential graded algebra thus the shuffle product endows the(ordinary) Hochschild complex CHlowast(Ω

lowastΩlowast) with the structure of a differentialgraded commutative algebra (cf [Lo92] Cor 427 p 125) On the other handwe have for a simply connected manifold M

26

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

Lemma 6 There is an quasi-isomorphism of commutative differential gradedalgebras

Ωlowast(LMUh) ≃ CHlowast(ΩlowastΩlowast)

Proof Let us first observe that the loop space with sitting instant LM = C ishomotopically equivalent to infin(S1M) A retraction is given by precomposingby a path in the circle with sitting instants and winding number one

Therefore our assertion is a version with coefficients in the graded associa-tive algebra Uh of Corollary 26 p 11 in [Lo11] originally shown by Chen[Ch73] Observe that the coefficients do not contribute to the differentials

In conclusion we obtain a homology class

[P (A0)] isin HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In the next section we will explain how to interprete

HHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

in terms of higher Hochschild homology as HHTlowast (Ω

lowastΩlowast) the higher Hochschildhomology of the 2-dimensional torus T We therefore obtain

[P (A0)] isin HHTlowast (Ω

lowastΩlowast)

7 Proof of the main theorem

In this section we consider higher Hochschild homology It has been introducedby Pirashvili in [Pi00] and further developed by Ginot Tradler and Zeinalian in[GTZ09] Here we follow closely [GTZ09]

In order to define higher Hochschild homology it is essential to restrictto commutative differential graded associative algebras Ωlowast We will see belowexplicitly why this is the case

Denote by the (standard) category whose objects are the finite orderedsets [k] = 0 1 k and morphisms f [k] rarr [l] are non-decreasing maps iefor i gt j one has f(i) ge f(j) Special non-decreasing maps are the injectionsδi [kminus1] rarr [k] characterized by missing i (for i = 0 k) and the surjectionsσj [k] rarr [k minus 1] which send j and j + 1 to j (equally for j = 0 k)

Denote by Setsfin the category of finite sets A finite simplicial set Ybull is bydefinition a contravariant functor Ybull op rarr Setsfin The sets of k-simplicesare denoted Yk = Y ([k]) The induced maps di = Ybull(δi) and sj = Ybull(σj) arecalled faces and degeneracies respectively Let Ybull be a pointed finite simplicialset For k ge 0 we put yk = |Yk|minus 1 ie one less than the cardinal of the finiteset Yk

The higher Hochschild chain complex of Ωlowast associated to the simplicial setYbull (and with values in Ωlowast) is defined by

CHYbullbull (ΩlowastΩlowast) =

983120

nisinZCHYbull

n (ΩlowastΩlowast)

27

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

In order to define the differential define induced maps as follows For any mapf Yk rarr Yl of pointed sets and any (homogeneous) element motimesa1otimes otimesayk

isinΩlowast otimes (Ωlowast)otimesyk we denote by flowast Ωlowast otimes (Ωlowast)otimesyk rarr Ωlowast otimes (Ωlowast)otimesyl the map

flowast(motimes a1 otimes otimes ayk) = (minus1)983171notimes b1 otimes otimes byl

where bj = Πiisinfminus1(j)ai (or bj = 1 in case fminus1(j) = empty) for j = 0 yl andn = m middot Πiisinfminus1(basepoint)j ∕=basepointai The sign 983171 is determined by the usualKoszul sign rule The above face and degeneracy maps di and sj induce thus

boundary maps (di)lowast CHYbullk (ΩlowastΩlowast) rarr CHkminus1 = kYbull(ΩlowastΩlowast) and degeneracy

maps (sj)lowast CHYbullkminus1(Ω

lowastΩlowast) rarr CHYbullk (ΩlowastΩlowast) which are once again denoted di

and sj by abuse of notation Using these the differential D CHYbullbull (ΩlowastΩlowast) rarr

CHYbullbull (ΩlowastΩlowast) is defined by setting D(a0 otimes a1 otimes otimes ayk

) equal to

yk983131

i=0

(minus1)k+983171ia0 otimes otimes diai otimes otimes ayk+

k983131

i=0

(minus1)idi(a0 otimes otimes ayk)

where 983171i is again a Koszul sign (see the explicit formula in [GTZ09]) Thesimplicial relations imply that D2 = 0 (this is the instance where one usesthat Ωlowast is graded commutative) These definitions extend by inductive limit toarbitrary (ie not necessarily finite) simplicial sets

The homology of CHYbullbull (ΩlowastΩlowast) with respect to the differentialD is by defini-

tion the higher Hochschild homology HHYbullbull (ΩlowastΩlowast) of Ωlowast associated to the sim-

plicial set Ybull In fact for two simplicial sets Ybull and Y primebull which have homeomorphic

geometric realization the complexes (CHYbullbull (ΩlowastΩlowast) D) and (CH

Y primebull

bull (ΩlowastΩlowast) D)are quasi-isomorphic thus the higher Hochschild homology does only dependon the topological space which is the realization of Ybull Therefore we will for ex-ample write HHT

bull (ΩlowastΩlowast) for the higher Hochschild homology of Ωlowast associated

to the 2-dimensional torus T inferring that it is computed with respect to somesimplicial set having T as its geometric realization

For the simplicial model of the circle S1 given in Example 231 in [GTZ09]one obtains the usual Hochschild homology In this sense HHYbull

bull generalizesordinary Hochschild homology

Example 245 in [GTZ09] givesFor the simplicial model of the 2-torus T given in Example 232 (of [GTZ09])

the algebra CHTbull (Ω

lowastΩlowast) is quasi-isomorphic to

CHbull(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast))

In this sense the holonomy cycle P (A0) (constructed in the previous section)may be regarded as living in the higher Hochschild complex CHT

bull (ΩlowastΩlowast) This

completes the proof of Theorem 1

Remark 7

Observe that the element P (A0) in CHTbull (Ω

lowastΩlowast) is of total degree zero Recall

28

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

from [GTZ09] (Corollary 247) the iterated integral map ItYbull of [GTZ09] whichprovides a morphism of differential graded algebras

ItYbull CHYbullbull (ΩlowastΩlowast) rarr Ωbull(MT Uh)

Here MT = Cinfin(TM) The image of P (A0) in Ωbull(MT Uh) represents a degreezero cohomology class which associates to each map f T rarr M an element ofUh which is interpreted as the gerbe holonomy taken over f(T) sub M We believethat an explicit expression of this cohomology class (in the special case of anabelian gerbe where all forms are real-valued) is given exactly by Gawedzki-Reisrsquo formula (214) [GaRe02] The factors gijk do not appear in our formulabecause we did not do the gluing yet and therefore everything is local

Observe further that following the steps in the proof of Corollary 244 of[GTZ09] one may express P (A0) in terms of matrices in A and B This is whatwe do in the next section

8 Explicit expression for the holonomy cycle

In order to find an explicit expression for the holonomy cycle we have totranslate first the connection 1-form A0 isin Ωlowast(LMUh) into an element ofCHlowast(Ω

lowastΩlowast) (with Ωlowast = Ωlowast(MUh)) using the quasi-isomorphism Ωlowast(LMUh) ≃CHlowast(Ω

lowastΩlowast) The second step is then to translate the holonomy cycle P (A0) isinCHlowast(CHlowast(Ω

lowastΩlowast) CHlowast(ΩlowastΩlowast)) using this expression of A0

The construction of A0 uses an iterated integral involving the 1-form A isinΩ1(M g) and the 2-form B isin Ω2(M h) Let us denote by 983144A0 the form 983144A0 isinΩ1(LMUg otimes Uh) which arises before combining the g coefficients of the A-components of the iterated integral with the h coefficient of the B-componentusing the action of g on h This form is in fact in Ω1(LMUg oplus h) because

the h coefficients arise only at one place namely in B The transcription of 983144A0

into an element of CHlowast(Ωlowast(M goplus h)Ωlowast(M goplus h)) is rather easy The explicit

expression is

983144A0 =

infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB

The form A0 is derived from this using two morphisms We use the factthat a Lie algebra homomorphism φ m rarr n induces a morphism of differ-ential graded Lie algebras φlowast Ωlowast(Mm) rarr Ωlowast(M n) The first homomor-phism of Lie algebras is the action α Ug rarr gl(h) sending x1 middot middot xr toα(x1α(x2 α(xrminus) )) This is well defined because α is zero on the idealgenerated by xotimes yminus yotimes xminus [x y] in the tensor algebra Tg on g and it is a Liealgebra homomorphism (using that g acts on h by derivations)

The second Lie algebra homomorphism is more involved It is the evaluationmorphism ev End(h)oplus h rarr h This is a Lie algebra homomorphism only if weconsider the zero bracket on h As we do have this restriction for other reasonsat one place we shall use it here also This explains how to obtain A0 from

29

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

983144A0 by applying two Lie algebra homomorphisms on the coefficient side Thisreasoning permits to split form part and coefficient part of A0

From the above expression for 983144A0 we obtain easily an expression for P ( 983144A0)

P ( 983144A0) =

infin983131

l=0

1⊠983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060⊠ ⊠

983059 infin983131

n=0

1otimesAotimes otimesA983167 983166983165 983168nminustimes

otimesB983060

983167 983166983165 983168lminustimes

This is the element in P ( 983144A0) isin CHlowast(CHlowast(ΩlowastΩlowast) CHlowast(Ω

lowastΩlowast)) whereΩlowast = Ωlowast(M goplus h) and where ⊠ denotes the tensor product in the Hochschild

complex as opposed to the tensor product otimes which occurs in 983144A0 before applyingthe above homomorphisms to obtain A0 In order to obtain an explicit expres-sion for P ( 983144A0) in the higher Hochschild complex with respect to the torus T2 wehave to apply the quasi-isomorphisms of Corollary 244 in [GTZ09] The resultwill be displayed in terms of matrices in accordance with the chain model forhigher Hochschild homology of the torus T2 given in Example 232 in [GTZ09]

As an example let us treat one of the most simple terms namely the form1⊠ (1otimes B) ⊠ (1otimes B) The first step is to apply degeneracies in order to havethe same degree in terms of otimesrsquos and ⊠rsquos This means we pass to

(1otimes 1otimes 1)⊠ (1otimes 1otimesB)⊠ (1otimesB otimes 1)

We will write simply 1 for 1 otimes 1 otimes 1 or 1 otimes 1 otimes 1 otimes 1 otimes 1 The second stepis to apply degeneracies according to shuffles In our example we have p = 2(internal S1) and q = 2 (external S1) so we sum over all (2 2)-shuffles The 6terms of the sum are thus (up to signs)

(1) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1

(2) 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ 1

(3) 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)

(4) 1⊠ 1⊠ (1otimes 1otimes 1otimes 1otimesB)⊠ (1otimesB otimes 1otimes 1otimes 1)⊠ 1

(5) 1⊠ 1⊠ (1otimes 1otimes 1otimesB otimes 1)⊠ 1⊠ (1otimesB otimes 1otimes 1otimes 1)

(6) 1⊠ 1⊠ 1⊠ (1otimes 1otimesB otimes 1otimes 1)⊠ (1otimesB otimes 1otimes 1otimes 1)

They correspond in this order to the shuffles (1 lt 2 3 lt 4) (1 lt 3 2 lt 4)(1 lt 4 2 lt 3) (2 lt 3 1 lt 4) (2 lt 4 1 lt 3) (3 lt 4 1 lt 2) The firstcomponent is p the second is q The shuffles indicate where to place the 1rsquos(internally and externally) These six terms may be translated into the followingmatrices

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 B 11 1 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 B1 1 1 1 11 1 B 1 11 1 1 1 1

983092

983110983110983110983110983108

30

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

983091

983109983109983109983109983107

1 1 1 1 11 1 1 B 11 1 1 1 11 1 1 1 11 1 B 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 B1 B 1 1 11 1 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 B 11 1 1 1 11 B 1 1 1

983092

983110983110983110983110983108

983091

983109983109983109983109983107

1 1 1 1 11 1 1 1 11 1 1 1 11 1 B 1 11 B 1 1 1

983092

983110983110983110983110983108

Observe that in these matrices only the first line and column are both inter-secting the diagonal and also filled with 1rsquos If another pair linecolumn hadthis property it would mean that the corresponding matrix is a degenerateHochschild chain A similar discussion holds for the general term in P ( 983144A0)

Observe that matrices of the same type arose in the work of Tradler Wilsonand Zeinalian [TWZ11] Remark 410 in the context of abelian gerbes

References

[ATZ10] H Abbaspour T Tradler M Zeinalian Algebraic String Bracket asa Poisson Bracket J Noncommut Geom 4 (2010) no 3 331ndash347

[AbZe07] H Abbaspour M Zeinalian String bracket and flat connections Al-gebr Geom Topol 7 (2007) 197ndash231

[BaCr04] J C Baez A S Crans Higher-Dimensional Algebra VI Lie 2-algebras Theory Appl Categ 12 (2004) 492ndash538

[BaSc04] J C Baez U Schreiber Higher Gauge Theory 2-Connections on 2-bundles arXivhep-th0412325 (this is not published only in abbreviatedform )

[Bar04] T Bartels 2-Bundles and Higher Gauge Theory Thesis(PhD)University of California Riverside 2006 142 pp

[Bre94] L Breen On the Classification of 2-Gerbes and 2-Stacks Asterisque225 (9) 1994

[BreMe] L Breen W Messing Differential geometry of gerbes Adv Math 198(2005) no 2 732ndash846

[CLS10] S Chatterjee A Lahiri A N Sengupta Parallel Transport Over PathSpaces Reviews in Math Phys 22 9 (2010) 1033ndash1059

[Ch73] K T Chen Iterated Integrals of differential forms and loop space ho-mology Ann Math (2) 97 (1973) 217ndash246

31

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

[GaRe02] K Gawedzki N Reis WZW branes and gerbes Rev Math Phys14 (2002) no 12 1281ndash1334

[Ge09] E Getzler Lie Theory for nilpotent Linfin-algebras Ann of Math (2) 170(2009) no 1 271ndash301

[Gi08] G Ginot Higher order Hochschild cohomology C R Math Acad SciParis 346 (2008) no 1-2 5ndash10

[GTZ09] G Ginot T Tradler M Zeinalian A Chen Model for Mapping Spacesand the Surface Product Ann Sci Ec Norm Super (4) 43 (2010) no 5811ndash881

[GiSt08] G Ginot M Stienon G-gerbes principal 2-group bundles and char-acteristic classes J Symplectic Geom 13 (2015) no 4 1001ndash1047

[Gi71] J Giraud Cohomologie non-abelienne Grundlehren der mathematischenWissenschaften in Einzeldarstellungen 197 1971

[He08] A Henriques Integrating Linfin algebras Compos Math 144 (2008) no4 1017ndash1045

[LSX10] C Laurent-Gengoux M Stienon P Xu Non-abelian differentiablegerbes Adv Math 220 (2009) no 5 1357ndash1427

[Lo82] J-L Loday Spaces with finitely many non-trivial Homotopy Groups JPure and Appl Algebra 24 (1982) 179ndash202

[Lo92] J-L Loday Cyclic Homology Grundlehren der mathematischen Wis-senschaften 301 (1992) Springer Verlag

[Lo11] J-L Loday Free loop space and homology Free loop spaces in geometryand topology 137ndash156 IRMA Lect Math Theor Phys 24 Eur MathSoc Zurich 2015

[LoKa82] J-L Loday C Kassel Extensions centrales drsquoalgebres de Lie AnnInst Fourier (Grenoble) 324 (1982) 119ndash142

[NiWa11] T Nikolaus K Waldorf Four Equivalent Versions of Non-AbelianGerbes Pacific J Math 264 (2013) no 2 355ndash419

[MaPi07] J F Martins R Picken On Two-Dimensional Holonomy TransAmer Math Soc 362 (2010) no 11 5657ndash5695

[Ne04] K-H Neeb Current groups for non-compact manifolds and their centralextensions Infinite dimensional groups and manifolds 109183 IRMA LectMath Theor Phys 5 de Gruyter Berlin 2004

[Ne07] K-H Neeb Non-abelian Extensions of Infinite-dimensional Lie groupsAnn Inst Fourier Grenoble 571 (2007) 209ndash271

32

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33

[Pi00] T Pirashvili Hodge decomposition for higher Hochschild homology AnnSci Ec Norm Super (4) 332 (2000) 151ndash179

[Ro07] D Roytenberg On weak Lie 2-algebras XXVIWorkshop on GeometricalMethods in Physics 180ndash198 AIP Conf Proc 956 Amer Inst PhysMelville NY 2007

[SSS09] H Sati U Schreiber J Stasheff Linfin-algebra connections and appli-cations to String- and Chern-Simons n-transport Quantum field theory303424 Birkhauser Basel 2009

[ScWa08a] U Schreiber K Waldorf Smooth Functors vs Differential FormsHomology Homotopy Appl 13 (2011) no 1 143ndash203

[ScWa08b] U Schreiber K Waldorf Connections on non-abelian gerbes andtheir holonomy Theory Appl Categ 28 (2013) 476ndash540

[Stash92] J Stasheff Differential graded Lie algebras quasi-Hopf algebras andhigher homotopy algebras in Quantum groups Number 1510 in LectureNotes in Math Springer Berlin 1992

[TWZ11] T Tradler S Wilson M Zeinalian Equivariant holonomy for bundlesand abelian gerbes Comm Math Phys 315 (2012) no 1 39ndash108

[CST04] J Cuntz G Skandalis B Tsygan Cyclic homology in non-commutative geometry Encyclopaedia of Mathematical Sciences 121Operator Algebras and Non-commutative Geometry II Springer-VerlagBerlin 2004

[Wa06] F Wagemann On Lie algebra crossed modules Comm Algebra 34 5(2006) 1699ndash1722

[WaWo11] F Wagemann C Wockel A Cocycle Model for Topological and LieGroup Cohomology Trans Amer Math Soc 367 (2015) no 3 1871ndash1909

[Wo09] C Wockel Principal 2-bundles and their gauge groups Forum Math23 (2011) no 3 565ndash610

33