carey wang

THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY

ALAN L. CAREY, BAI-LING WANG

Abstract. The goal of this paper is to apply the universal gerbe developed in [CMi1] and

[CMi2] and the local family index theorems to give a unified viewpoint on the known examples

of geometrically interesting gerbes, including the determinant bundle gerbes in [CMMi1],

the index gerbe in [L] for a family of Dirac operators on odd dimensional closed manifolds.

We also discuss the associated gerbes for a family of Dirac operators on odd dimensional

manifolds with boundary, and for a pair of Melrose-Piazza’s Cl(1)-spectral sections for a

family of Dirac operators on even dimensional closed manifolds with vanishing index in K-

theory. The common feature of these bundle gerbes is that there exists a canonical bundle

gerbe connection whose curving is given by the degree 2 part of the even eta-form (up to an

locally defined exact form) arising from the local family index theorem.

1. Introduction

Bundle gerbes on differentiable manifolds are introduced by Murray in [Mur] as differential

geometric objects corresponding to Brylinski’s description in terms of sheaves of groupoids for

Giraud’s gerbes in [Bry].

Associated with a smooth submersion π : Y → M , let Y [2] the fiber product of π with the

obvious groupoid structure. A bundle gerbe G over M , as defined in [Mur], consists of a sujective

submersion π : Y → M , and a principal U(1)-bundle G over Y [2] = Y ×π Y together with a

groupoid multiplication on G, which is compatible with the natural groupoid multiplication on

Y [2]. We represent a bundle gerbe G over M as the following diagram

G

Y [2]π1

//

π2

// Y

M

(1)

with the bundle gerbe product given by an isomorphism

m : π∗1G ⊗ π∗

3G⊗ → π∗2G(2)

as principal U(1)-bundles over Y [3] = Y ×πY ×πY , and πi, i=1,2, 3, are three natural projection

from Y [3] to Y [2] by omitting the entry in i position for πi. The tautological bundle gerbes

in terms of path fibrations and the lifting bundle gerbes associated to a central extension of

groups are given as examples in [Mur].

The bundle gerbe in its stable isomorphism class admits a canonical isomorphism class given

by its local systems: the local bundle gerbe (Cf. [MS]). When M is covered by a good cover

Uα such that all their finite intersections are contractible and the submersion π admits a1

2 ALAN L. CAREY, BAI-LING WANG

local section sα over Uα, then the local bundle gerbe can be described by a family of local

U(1)-bundles Gαβ over Uαβ = Uα ∩ Uβ, the pull-back of G via (sα, sβ):⊔

αβ Gαβ

⊔

αβ Uαβ

π1//

π2

//⊔

α Uα

M

(3)

and the bundle gerbe product (2) is given by an isomorphism

φαβγ : Gαβ ⊗ Gβγ∼= Gαγ ,(4)

over Uαβγ , such that φαβγ is associative as a groupoid multiplication. A related local picture

of gerbes has been introduced by Hitchen [Hit].

A Cech cocycle fαβγ can be obtained from the isomorphism (4) by choosing a section sαβ

of Gαβ , i.e.,

φαβγ(sαβ ⊗ sβγ) = fαβγ · sαγ

for a U(1)-valued function fαβγ over Uαβγ . The equivalence class of fαβγ doesn’t depend on

the choices of local section sαβ, and represents the Dixmier-Douady class of the bundle gerbe

in

H2Cech(M, U(1)) ∼= H3(M, Z),

where U(1) is the sheaf of continuous U(1)-valued functions over M .

The geometry of bundle gerbes, bundle gerbe connections and their curvings are studied

in [Mur]. On the corresponding local bundle gerbe (3), a bundle gerbe connection is a fam-

ily of U(1)-connections ∇αβ on Gαβ which is compatible with the isomorphism (4), i.e.,

Aαβ = ∇αβsαβ/sαβ satisfies

Aαβ + Aβγ + Aγα = f−1αβγdfαβγ .(5)

A curving then is a locally defined 2-form Bα, the so-called B-field, such that,

Bα − Bβ = dAαβ ,(6)

over Uα ∩ Uβ. We remark that the B-field Bαα is unique, up to locally defined exact 2-

forms. For the B-field Bα + dCαα, then we need to modify the gerbe connection by adding

Cα−Cβ to Aαβ without any effect on the compatibility condition (5). This local description

of connection and curving for the canonical local bundle gerbe defines an element

[(fαβγ , Aαβ , Bα)]

in the degree three Deligne cohomology group H3Del(M,D3). The curvature of B-field is given

by dBα, note that

dBα = dBβ , on Uαβ

is a globally defined 3-form which represents the image of the Dixmier-Douady class in H3(M, R)

up to a factor ofi

2π, just like the case of line bundles with connections. In this paper, we

often suppress this factor and identify the bundle gerbe curvature with the differential form

representing the image of the Dixmier-Douady class in H3(M, R).

THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 3

To be precise, a B-field of a bundle gerbe G over M should be a Hopkins-Singer differential

cocycle as defined in [HS] whose equivalence class is the corresponding degree three Deligne

cohomology class. Recall that the following commutative diagram

H3Del(M,D3)

// H3(M, Z)

Ω3Z(M) // H3(M, R)

where Ω3Z(M) is the space of closed 3-form on M with integral periods. In this sense, we often

say that the curvature of the B-field represents the image of the Dixmier-Douady class of the

corresponding bundle gerbe in H3(M, R), even loosely speaking, the curvature of the B-field

represents the Dixmier-Douady class of the corresponding bundle gerbe.

We remark that there is a subtlety here relating to the gauge transformations of the B-field.

To illustrate this subtlety, given a globally defined closed 2-form ω (not necessarily of integer

periods), with respect to a cover Uα of M we can write ω|Uα = dθα as an exact 2-form by

Poincare Lemma. Then on the one hand, one can see that

(1, 0, ω|Uα)

represents a trivial degree three Deligne class in H3Del(M,D3) if and only if the closed 2-form

ω has integer periods. This claim follows from the exact sequence

0 −→ Ω2(M)/Ω2Z(M) −→ H3(M,D3) −→ H3(M, Z) −→ 0.

In this sense, the gauge transformation of the B-field is given by line bundles with connection

on M . On the other hand,

(1, θα − θβ, ω|Uα) = (1, θα − θβ , dθα)

always represents a trivial degree three Deligne class in H3Del(M, Z), as (1, θα − θβ, dθα) is a

coboundary element. In this paper, a curving on a bundle gerbe is always defined up to a locally

defined exact 2-form in this latter sense so that the corresponding B-field is well-defined up to a

degree three Deligne coboundary term of form (1, θα − θβ, dθα), here dθα may not be a globally

defined exact 2-form. We hope that this remark will clarify any confusion when we talk about

curvings on a bundle gerbe, such as degree 2 part of the even eta-form up to an exact 2-form.

Earlier applications to various aspects of quantum field theory are extensively exploited in

[BoMa][BCMMS] [CMu1][CMMi1][CMMi2] [CMi1][CMi2] [EM][GR] [Mic] which cover a wide

range of applications. For example they provide intrinsic geometric meaning to string struc-

tures on manifolds (a problem originating from string theory), give a topological sense to local

Hamiltonian or commutator anomalies (also referred to as Schwinger terms) for the gauge

group action on fermionic Fock spaces, describe the geometry of Wess-Zumino-Witten actions

and finally give a framework for the study of twisted K-theory for D-brane charges.

A significant application comes from the construction of the universal gerbe in [CMi1][CMi2]

where Carey and Mickelsson analysed the obstruction to obtaining a second quantization for a

smooth family of Dirac operators on an odd dimensional spin manifold. Explicit computations

of the Dixmier-Douady class for the universal gerbe were also done in [CMi1][CMi2].

Using techniques from local family index theory, in [L], Lott gives a construction of the higher

degree analogs of the determinant line bundle for a family of Dirac operators on odd-dimensional


closed spin manifolds, which realized Carey and Mickelsson’s universal gerbe in [CMi1][CMi2]

in this particular case. The connective structures and gerbe holonomy are discussed by Bunke

in [Bu1]. The construction in Lott’s paper [L] is rather technical and doesn’t work for a family

of Dirac operators on odd-dimensional manifolds with boundary.

In the present paper, we will apply the universal gerbe developed in [CMi1] [CMi2] to give a

unified viewpoint on the known examples of geometrically interesting gerbes and also construct

gerbes for a family of Dirac operators on odd dimensional manifolds with boundary and for a pair

of Melrose-Piazza Cl(1)-spectral sections for a family of Dirac operators on even dimensional

closed manifolds with vanishing index in K-theory. These latter gerbes exhaust all bundle

gerbes corresponding to the Dixmier-Douady classes in the image of the Chern character map

on the K1-group of the underlying manifold. In all these examples of gerbes, there exists a

canonical bundle gerbe connection whose curving, up to an exact 2-form, is given by the degree

2 part of the even eta-form arising from the local family index theorem.

The framework for our study of the geometry of bundle gerbes in this paper is provided

by the four famous local family index theorems developed by Bismut[Bis2], Bismut-Freed[BF],

Bismut-Cheeger[BC2] and Melrose-Piazza [MP1][MP2], which are the corner-stones of local

family index theory. For the convenience of readers and to set up various notations, we give a

brief review of local family index theory in Section 2, and state these four local family index

theorems as Theorems 2.2, 2.4, 2.6 and 2.7.

In Section 3, we recall the construction of the determinant bundle gerbe in [CMMi1] and

study its geometry in Theorem 3.1.

In Section 4, we give a complete proof of the existence theorem (Theorem 4.1) of the canonical

bundle gerbe associated to any element in K1-group using the universal gerbe developed in

[CMi1][CMi2].

In Section 5, we apply the universal gerbe and the canonical bundle gerbe constructed in

section 4 to discuss the index gerbe associated to a family of Dirac operators on odd dimensional

manifold with or without boundary. This gives a simple construction of the gerbe, which was

discussed in [L] and [Bu1], for a family of Dirac operators on an odd dimensional closed manifold.

Two new examples of bundle gerbes are obtained, one for a family of Dirac operators on odd

dimensional manifolds with boundary, and the other for a pair of Melrose-Piazza Cl(1)-spectral

sections for a family of Dirac operators on even dimensional closed manifolds with vanishing

index in K-theory. Some relations among these bundle gerbes are also briefly included at the

end of this section.

Our descriptions of the universal gerbe and its applications demonstrate on the one hand

that these powerful local family index theorems provide simple and clear perspectives to study

those bundle gerbes associated to families of Dirac operators, and on the other hand that the

bundle gerbe theory also motivates the further understanding of local family index theory for

manifolds with corners. For a single manifold with corners, Fredholm perturbations of the

Dirac type operators and their index are throughly studied by Loya and Melrose in [LM]. A

different approach to these latter questions is contained in [Bu2]. We intend the present paper

as a different and much more accessible method for tackling these issues.

Acknowledgement ALC acknowledges advice from P. Piazza on spectral sections and also

thanks V. Mathai for conversations on his joint work with R. Melrose related to the theme


analysed in this paper. BLW wishes to thank Xiaonan Ma for many invaluable comments and

his explanations of eta forms.

2. Review of local family index theory

In this section, we briefly review local index theory and eta forms. Local family index theo-

rems refine the cohomological family index theorem of Atiyah-Singer, while eta forms transgress

the local family index theorem at the level of differential forms. The main reference for this

section is Bismut’s survey paper [Bis1] and the references therein.

Let π : X → B be a smooth fibration over a closed smooth manifold B, whose fibers are

diffeomorphic to a compact, oriented, spin manifold M . Let (V, hV ,∇V ) be an Hermitian vector

bundle over X equipped with a unitary connection.

Let gX/B be a metric on the relative tangent bundle T (X/B) (the vertical tangent subbundle

of TX), which is fiberwise a product metric near the boundary if the boundary of M is non-

empty. In the latter case, ∂X → B is also a fibration with compact fiber ∂M . Let SX/B be the

corresponding spinor bundle associated the spin structure on (T (X/B), gX/B).

Let T HX be a smooth vector subbundle of TX , (a horizontal vector subbundle of TX), such

that

TX = T HX ⊕ T (X/B),(7)

and such that if ∂M is non-empty, T HX |∂X ⊂ T (∂X). Then T H(∂X) = T HX |∂X is a hori-

zontal subbundle of T (∂X).

Denote by P v the projection of TX onto the vertical tangent bundle under the decomposi-

tion (7). As shown in Theorem 1.9 of [Bis2], (T HX, gX/B) determines a canonical Euclidean

connection ∇X/B as follows. Choose a metric gX on TX such that T HX is orthogonal to

T (X/B) and such that gX/B is the restriction of gX to T (X/B). Then

∇X/B = P v ∇X

where ∇X is the Levi-Civita connection on (TX, gX).

Write

A(x) =x/2

sinh(x/2),

Ch(x) = exp(x).

Then the corresponding characteristic classes can be represented by closed differential forms on

X as

A(T (X/B),∇X/B) = det1/2(

A(i(∇X/B)2

2π))

,

Ch(V,∇V ) = Tr(

Ch(i(∇V )2

2π))

.

For any b ∈ B, let /Db be the Dirac operator acting on C∞(Xb, (SX/B ⊗ V )|Xb), where

Xb = π−1(b) is the fiber of π over b. Then /Dbb∈B is a family of elliptic operators over B

acting on an infinite dimensional vector bundle H where

Hb = C∞(Xb, (SX/B ⊗ V )|Xb

).

The Hermitian metrics on the spinor bundle SX/B and V induce a natural L2 Hermitian

metric on H , as we can identify smooth sections of H with smooth sections of SX/B ⊗ V ) on

X and integrate along the fiber using the Riemannian volume form.


Denote by ξ the horizontal lift of a vector field ξ ∈ TB. Define

T (ξ1, ξ2) = −P v([ξ1, ξ2]).

Let fi be a local frame of TB, and let f i be the dual frame of T ∗B. Set

c(T ) =1

2f if jc(T (fi, fj)),

where c(·) is the Clifford action of T (X/B) on SX/B . Then c(T ) ∈ Λ(T ∗B)⊗End(H), the

graded tensor product when H is Z2-graded in the case of even fibers.

Define a connection ∇H on H by

∇Hξ s = ∇SX/B⊗V

ξs,

where ∇SX/B⊗V is the induced connection on SX/B ⊗ V from ∇X/B and ∇V , ξ ∈ TB, and s is

a smooth section of SX/B ⊗ V on X .

Let dvolM be the Riemannian volume form along the fibers M , the Lie derivative operator

Lξ acting on tensors along the fiber M defines a tensor

divM (ξ) =Lξ(dvolM )

dvolM.

Then ∇H,uξ = ∇H

ξ +1

2divM (ξ) defines a unitary connection on H (cf. [Bis2]). The curvature of

∇H,uξ takes values in first order differential operators acting along the fibers of π.

For convenience in the expression of local index formulae, let φ be the endomorphism of

Λ(T ∗B) such that φ(ω) = (2πi)−degω/2ω.

2.1. Local family index theorem for closed manifolds. Case 1: Even dimensional

fibers

We assume that the fibers of π : X → B are even dimensional closed manifolds. Then the

spinor bundle SX/B = S+X/B ⊕ S−

X/B is Z2-graded, hence, H = H+ ⊕ H− with

H± = C∞(Xb, (S±X/B ⊗ V )|Xb

)

and /Dbb∈B is a family of self-adjoint, elliptic odd operators over B interchanging

C∞(Xb, (S+X/B ⊗ V )|Xb

) and C∞(Xb, (S−X/B ⊗ V )|Xb

). Let /D±,b be the restriction of /Db to

C∞(Xb, (S±X/B ⊗ V )|Xb

), so

/Db =

(

0 /D−,b

/D+,b 0

)

Then the family /Dbb∈B defines a K0-element

Ind( /D+) ∈ K0(B).

The Atiyah-Singer family index formula is given by

Ch(

Ind( /D+))

= π∗[A(T (X/B))Ch(V )] ∈ Heven(B, R)(8)

where π∗ the push-forward map given by the integration along the fibers on the level of differ-

ential forms.

In order to get explicit differential forms, we need to introduce a Bismut superconnection

for Z2 graded vector bundles. Let E = E+ ⊕ E− be a Z2 graded complex vector space, let


τ = ±1 on E± defining the Z2-grading. The supertrace for a trace class operator A ∈ End(E)

is defined to be

Trs(A) = Tr(τA).

Note that the supertrace vanishes on any supercommutator.

For a Hermitian Z2 vector bundle E with a unitary connection ∇E preserving the grading,

then the super Chern character for (E,∇E) is defined to be

Chs(E,∇E) = Trs

(

Ch(i(∇E)2

2π))

.

The Bismut superconnection on the infinite dimensional Z2-graded bundle H , adapted to /D,

is defined to be

At = ∇H,u +√

t /D − c(T )

4√

t,(9)

acting on Ω∗(B)⊗C∞(B, H). Note that the fiberwise ellipticity of At implies that the heat

operator

exp(−A2t )

is fiberwise trace class in the supertrace sense.

Definition 2.1. Define

αt = φTrs

(

exp(−A2t ))

,

βt =1√2πi

φTrs

(∂At

∂texp(−A2

t ))

.(10)

The following local family index theorem of ([Bis2] and [BGV]) refines the Atiyah-Singer

family index formulae (8),

Theorem 2.2. The forms αt and βt are real and satisfy∂αt

∂t= −dβt. The form αt is closed

and its cohomology class [αt] is constant and

[αt] = Ch(Ind( /D+)) ∈ Heven(B, R).

Moreover αt and βt have the following asymptotic behaviours.

(a) As t → 0,

αt = π∗

(

A(T (X/B),∇X/B)Ch(V,∇V ))

+ O(t),

βt = O(1).

(b) Assume that Ker /D is a smooth subbundle of H with ∇Ker /D,u given by the orthogonal

projection of ∇H,u on Ker /D. As t → ∞,

αt = Chs(Ker /D,∇Ker /D,u) + O(t−12 ),

βt = O(t−32 ).

Define the odd eta form on B to be

ηodd =

∫ ∞

0

βtdt.

Then this odd eta form ηodd satisfies the transgression formula

dηodd = π∗

(


− Chs(Ker /D,∇Ker /D,u).(11)


Case 2: odd dimensional fibers

When M is an odd dimensional closed manifold, the family /Dbb∈B is a family of self-adjoint

operators on H . Then the Atiyah-Singer index theory tells us that

Ind( /D) ∈ K1(B)

with its odd Chern character given by

Ch(Ind( /D)) = π∗[A(T (X/B))Ch(V )] ∈ Hodd(B, R).(12)

To define the Bismut superconnection in this setting, we need to introduce an odd variable

such that σ2 = 1. We may do this by tensoring on a complex vector space on which the Clifford

element σ acts as well as a Z2-grading. Then for any complex vector bundle E on B, E ⊗C[σ]

and End(E) ⊗ C[σ] are Z2-graded in the obvious way. Define the σ-trace to be the functional

Λ(T ∗B)⊗(

End(E) ⊗ C[σ])

→ Λ(T ∗B),

such that if ω ∈ Λ(T ∗B), and A ∈ End(E), then

Trσ(ωA) = 0 Trσ(ωAσ) = ωTr(A).

Define the Bismut superconnection for the odd dimensional case to be

At = ∇H,u +√

t /Dσ − c(T )σ

4√

t.(13)

Definition 2.3. Define

αt = (2i)12 φTrσ

(

exp(−A2t ))

,

βt =1√π

φTrσ(∂At

∂texp(−A2

t ))

.(14)

Then we have the following local family index theorem for the odd case ([BF]).

Theorem 2.4. The forms αt and βt are real and satisfy∂αt

∂t= −dβt. The form αt is closed,

its cohomology class [αt] is constant and

[αt] = Ch(Ind( /D)) ∈ Hodd(B, R).

Moreover αt and βt have the following asymptotic behaviours.

(a) As t → 0,

αt = π∗

(


+ O(t),

βt = O(1).

(b) Assume that Ker /D is a smooth subbundle of H. As t → ∞,

αt = O(t−12 ),

βt = O(t−32 ).

Define the even eta form on B to be

ηeven =

∫ ∞

0

βtdt.

Then this even eta form ηeven satisfies the following transgression formula

dηeven = π∗

(


.(15)


2.2. Local family index theorem for manifolds with boundary. Case 1: Even dimen-

sional fibers.

Assume that M is an even dimensional manifold with boundary ∂M . The family of boundary

Dirac operators /D∂Mb b∈B for the odd dimensional fibration ∂X → B, if Ker /D is a smooth

bundle over B, assigns an even eta-form ηeven on B satisfying the transgression formula (15)

dηeven =

∫

∂M

A(T (X/B),∇X/B)Ch(V,∇V ).

With the assumption of invertibility of the boundary Dirac operators, Bismut and Cheeger

proved the local family index theory in [BC2].

Theorem 2.5. Assume that Ker /D∂Mb = 0 for any b ∈ B. With the Atiyah-Patodi-

Singer boundary condition, the family of Dirac operators /DM+,≥ has a well-defined index

Ind( /DM+,≥) ∈ K0(B), whose Chern character is given by

Ch(

Ind( /DM+,≥)

)

=

∫

M

A(T (X/B),∇X/B)Ch(V,∇V ) − ηeven ∈ Heven(B, R).

Recall that the Atiyah-Patodi-Singer boundary condition is given by the family of orthogonal

projection operators P≥,b on the direct sum of the eigenspaces of /D∂Mb associated to the non-

negative spectrum for any b ∈ B. The invertibility condition ensures that P≥,b is a smooth

family of boundary conditions. The Dirac operator /DM+,≥;b acts on the space of L2

1 sections of

(S+X/B ⊗ V )|Xb

with boundary component belonging to KerP≥,b. Then /DM+,≥ is a family of

Fredholm operators.

For the general situation, Melrose-Piazza introduced the notion of a spectral section for the

family of boundary Dirac operators /D∂Mb b∈B in [MP1]. A spectral section is a smooth family

of self-adjoint pseudo-differential projections

Pb : C∞(∂Xb, (S∂M/B ⊗ V )|∂Xb) → C∞(∂Xb, (S∂M/B ⊗ V )|∂Xb

),

such that there is R > 0 for which /D∂Mb s = λs implies that

Pbs = s λ > R,

Pbs = 0 λ < −R.

The spectral section exists if and only if Ind( /D∂M ) = 0 ∈ K1(B). In our case, Ind( /D∂M ) = 0

follows from the cobordism invariance of the Atiyah-Singer index theory.

For a spectral section P , Melrose-Piazza also constructed a similar Bismut superconnection

(13) where /D∂M is replaced by an interpolator between /D∂M for t << 1 and a suitable pertur-

bation /D∂M + AP (with AP depending on P ) for t >> 1. The family AP,bb∈B is a family of

self-adjoint smooth operators such that /D∂Mb + AP,b is invertible. Then an even eta form ηP

even

can be defined as in Theorem 2.4 , modulo exact forms, depending only on /D∂Mb b∈B and

the spectral section P . They established the local family index theorem for even dimensional

manifolds with nonempty boundary ([MP1]) generalizing the Bismut-Cheeger Theorem 2.5.

Theorem 2.6. Let π : X → B be a smooth fibration with fiber diffeomorphic to an even

dimensional spin manifold M with boundary and the geometric data given as in the beginning

of this section. With the boundary condition given by a spectral section P for /D∂Mb b∈B, the


family of Dirac operators /DM+,P ;bb∈B has a well-defined index Ind( /DM

+,P ) ∈ K0(B), with the

Chern character form given by

Ch(

Ind( /DM+,P )

)

=

∫

M

A(T (X/B),∇X/B)Ch(V,∇V ) − ηPeven ∈ Heven(B, R).

Case 2: odd dimensional fibers

Now we assume that the fibers of the fibration π : X → B are odd dimensional manifolds

with non-empty boundary. The family of boundary Dirac operator /D∂Mb b∈B consists of self-

adjoint, elliptic, Z2-graded odd differential operators acting on

H = C∞(Xb, (S+X/B ⊗ V )|Xb

) ⊕ C∞(Xb, (S−X/B ⊗ V )|Xb

).(16)

The odd dimensional Dirac operator /DMb acting on C∞(Xb, (SX/B ⊗ V )|Xb

) is neither Fred-

holm nor self-adjoint without suitable boundary conditions. The Fredholm boundary conditions

as developed in [Sco0] form the restricted smooth Grassmannian with respect to the chirality

polarization (16). To get self-adjointness, we need to impose a Cl(1)-condition (where Cl(1)

denote the Clifford algebra Cl(R)):

c(du)PW + PW c(du) = c(du),

with u being the inward normal coordinate near the boundary ∂M , and PW is the orthogonal

projection corresponding to the Fredholm boundary condition W in the restricted smooth

Grassmannian.

In order to get a continuous family of self-adjoint Fredholm extensions from /DMb b∈B,

Melrose and Piazza introduced a Cl(1)-spectral section P and constructed an odd eta form

ηPodd which, modulo exact forms, depends only on /D∂M

b b∈B and the Cl(1)-spectral section

([MP2]).

Theorem 2.7. ([MP2]) Let π : X → B be a smooth fibration with fiber diffeomorphic to an odd

dimensional spin manifold M with boundary and the geometric data given as in the beginning

of this section. With the boundary condition given by a Cl(1)-spectral section P for /D∂Mb b∈B,

the family of Dirac operator /DMP ;bb∈B has a well-defined index

Ind( /DMP ) ∈ K1(B),

with the Chern character form given by

Ch(

Ind( /DMP ))

=

∫

M

A(T (X/B),∇X/B)Ch(V,∇V ) − ηPodd ∈ Hodd(B, R).(17)

2.3. Determinant line bundle. Let π : X → B be a smooth fibration over a smooth manifold

B, whose fibers are diffeomorphic to a compact, oriented, even dimensional spin manifold M .

Let (V, hV ,∇V ) be a Hermitian vector bundle over X equipped with a unitary connection.

Let gX/B be a metric on the relative tangent bundle T (X/B) (the vertical tangent subbundle

of TX), which is fiberwisely a product metric near the boundary SX/B be the corresponding

spinor bundle associated the spin structure on (T (X/B), gX/B).

With a choice of spectral section P for the family of boundary Dirac operators /D∂Mb b∈B,

we have a family of Fredholm operators /DM+,P ;bb∈B over B. There is a honest determinant

line bundle, denoted by Det( /DM+,P ), over B given by

Det( /DM+,P )b = Λtop(Ker /DM

+,P ;b)∗ ⊗ Λtop(Coker /DM

+,P ;b).


Using zeta determinant regularization, as in [Qu][BF][Woj][SW], a Hermitian metric and a

unitary connection ∇ /DM+,P can be constructed on Det( /DM

+,P ) such that

c1(Det( /DM+,P ),∇ /DM

+,P ) =[

π∗

(


− ηPeven

]

(2),(18)

Note that the eta form ηPeven is defined modulo exact forms, the above equality (18) holds

modulo exact 2-forms.

For a smooth fibration X over a smooth manifold B, with closed even-dimensional fibers

partitioned into two codimensional 0 submanifolds M = M0 ∪ M1 along a codimension 1

submanifold ∂M0 = −∂M1. Assume that the metric gX/B is of product type near the collar

neighborhood of the separating submanifold. Let P be a spectral section for for /D∂M0

b b∈B,

then I − P is a spectral section for /D∂M1

b b∈B. Scott showed ([Sco]) that the determinant

line bundles for these three families of /DM+;bb∈B, /DM0

+,P ;bb∈B and /DM1

+,I−P ;bb∈B satisfy the

following gluing formulae as Hermitian line bundles:

Det( /DM+ ) ∼= Det( /DM0

+,P ) ⊗ Det( /DM1

+,I−P ),(19)

moreover the splitting formulae for the curvature of ∇ /DM+ implies that

c1(Det( /DM+ ),∇ /DM

+ ) = c1(Det( /DM0

+,P ),∇ /DM0+,P ) + c1(Det( /DM1

+,I−P ),∇ /DM1+,I−P ).(20)

See ([MaP][Pi1] [Pi2]) for another explicit expressions of the corresponding Quillen metrics

and their unitary connections on Det( /DM+ ), Det( /DM0

+,P ) and Det( /DM1

+,I−P ), which are called the

b-Quillen metrics and the b-Bismut-Freed connections.

3. Determinant bundle gerbe

In [CMMi1], Carey-Mickelson-Murray applied the theory of bundle gerbes to study the bun-

dle of fermionic Fock spaces parametrized by vector potentials on odd dimensional manifolds

and explained the appearance of the so-called Schwinger terms for the gauge group action.

We briefly review this construction here. Let M be a closed, oriented, spin manifold with

a Riemannian metric gM . Let S be the spinor bundle over M and let (V, hV ) be a Hermitian

vector bundle.

Denote by A the space of unitary connections on V and by G the based gauge transformation

group, that is, those gauge transformations fixing the fiber over some fixed point in M . With

proper regularity on connections, the quotient space A/G is a smooth Frechet manifold.

Carey-Mickelson-Murray constructed a non-trivial bundle gerbe on A/G, and gave a formula

for its Dixmier-Douady class.

We summarise their construction in terms of the local family index theorem.

Let λ ∈ R, denote

Uλ = A ∈ A|λ /∈ spec( /DA),where the Dirac operator /DA acts on C∞(M, S ⊗ V ). Let H be the space of square integrable

sections of S ⊗ V . For any A ∈ Uλ, there is a uniform polarization:

H = H+(A, λ) ⊕ H−(A, λ)

given by the spectral decomposition of H with respect to /DA −λ into the positive and negative

eigenspaces. Denote by PH±(A,λ) the orthogonal projection onto H±(A, λ).

Fix λ0 ∈ R and a reference connection A0 ∈ A such that the Dirac operator /DA0− λ0 is

invertible, denote by PH±(A0,λ0) the orthogonal projection onto H±(A0, λ0).


Over Uλ, there exists a complex line bundle Detλ, which is essentially the determinant

line bundle for the even dimensional Dirac operators on [0, 1] × M coupled to some path of

connections in A with respect to the generalized Atiyah-Singer-Patodi boundary condition

([APS]). The path of vector potentials A(t) can chosen to be any smooth path connecting A0

and A ∈ Uλ, for simplicity, we take

A = A(t) = f(t)A + (1 − f(t))A0(21)

for t ∈ [0, 1] where f is zero on [0, 1/4], equal to one on [3/4, 1] and interpolates smoothly

between these values on [1/4, 3/4]. The generalized Atiyah-Singer-Patodi boundary condition

is determined by the orthogonal spectrum projection

Pλ = PH+(A0,λ0) ⊕ PH−(A,λ),

that is, the even dimensional Dirac operator on [0, 1] × M acts on the spinor fields whose

boundary components belong to H−(A0, λ0) at 0 × M and H+(A, λ) at 1 × M .

Note that Pλ can be thought as a spectral section in the sense of Melrose-Piazza [MP1] for

the family of Dirac operators

/DA0⊕ /DAA∈Uλ

,

on 0 × M ⊔ 1 × M . Then the eta form ηPλeven associated with /DA0

⊕ /DAA∈Uλand the

spectral section Pλ is an even differential form on Uλ, which is unique up to an exact form.

Denote by /DPλ

Athe even dimensional Dirac operator /DA with respect to the APS-type

boundary condition Pλ. Then the even dimensional Dirac operator

/DPλ

A=

(

0 /DPλ

A,−

/DPλ

A,+ 0

)

is a Fredholm operator, whose determinant line is given by

Detλ(A) = Λtop(Ker /DPλ

A,+)∗ ⊗ Λtop(Coker /DPλ

A,+).

The family of Fredholm operators /DPλ

A, parametrized by A ∈ Uλ defines a determinant

line bundle over Uλ, which is given by

Detλ =⋃

A∈Uλ

Detλ(A).

The determinant line bundle Detλ can be equipped with a Quillen metric and a Bismut-Freed

unitary connection whose curvature can be calculated by the local family index theorem.

Over Uλλ′ = Uλ ∩ Uλ′ , there exists a complex line bundle Detλλ′ such that

Detλλ′ = Det∗λ ⊗ Detλ′ .

These local line bundles Detλλ′ over A, form a bundle gerbe as established in [CMMi1].

In the following, we will apply the local family index theorem to study the geometry of this

determinant bundle gerbe. To apply the local family index theorem, we should restrict ourselves

to a smooth finite dimensional submanifold of A. For convenience however, we will formally

work on the infinite dimensional manifold A directly.

Consider the trivial fibration [0, 1]×M ×A over A with fiber [0, 1]×M an even dimensional

manifold with boundary. Over [0, 1] × M × A, there is a Hermitian vector bundle V which is

the pull-back bundle of V .


There is a universal unitary connection on V, also denoted by A, whose vector potential at

(t, x, A) is given by

A(t, x) = A(t)(x),(22)

where A(t) is given by (21). Denote by Ch(V, A) the Chern character of (V, A).

Now we can state the following theorem regarding the geometry of the bundle gerbe over

A/G constructed in [CMMi1].

Theorem 3.1. The local line bundles Detλλ′ descend to local line bundle over A/G, which

in turn defines a local bundle gerbe over A/G, the corresponding Dixmier-Douady class is give

by(

∫

M

A(TM,∇TM )Ch(V, A))

(3).

Moreover, the induced unitary connection and the even eta form ηPλeven (up to an exact 2-form)

descend to a bundle gerbe connection and curving on the local bundle gerbe over A/G

Proof. Equip Detλ with the Quillen metric and its unitary connection, then its first Chern

class is represented by the degree 2 part of the following differential form:∫

[0,1]×M

A(TM,∇TM )Ch(V, A) − ηPλeven.

In this formula ∇TM is the Levi-Civita connection on (TM, gM ), A(TM,∇TM ) represents the

A-genus of M , and ηPλeven is the even eta form on Uλ associated to the family of boundary

Dirac operators and the spectral section Pλ. Note that ηPλeven (modulo exact forms) is uniquely

determined, see Theorem 2.6.

The induced connection on Detλλ′ implies that its first Chern class is given by

(

ηPλeven − ηPλ′

even

)

(2).

Note that the eta form ηPλeven is unique up to an exact form (in a way analogous to the B-field),

hence,(

ηPλeven − η

Pλ′

even

)

(2)is a well-defined element in H2(M, R). From the transgression formula

for the eta forms and the Stokes formula, we know that over Uλλ′ ,

d(ηPλeven)(2) = d(ηPλ′

even)(2) =(

∫

M

A(TM,∇TM )Ch(V, A))

(3).(23)

Here use the fact that the contribution from 0 × M vanishes as a differential form on A, as

we fixed a connection A0 over 0 × M . We also use the same notation (V, A) to denote the

Hermitian vector bundle over M ×A and the universal unitary connection A.

As the gauge group acts on A and Detλ covariantly, by quotienting out G, we obtain the

bundle gerbe over A/G described in the theorem.

Remark 3.2. The Dixmier-Douady class in Theorem 3.1 is often non-trivial. For example

note that when dimM < 4, there is no contribution from the A-genus and in particular, for

dimM = 1 or 3, using the Chern-Simons forms, non-triviality was proved by explicit calculation

in [CMMi1].


The above construction can be generalized to the fibration case as in the local family index

theorem for odd dimensional manifolds with boundary as in section 2. When the fibers are

closed odd dimensional spin manifolds, this leads to the index gerbe as discussed by Lott ([L]).

In the next section, we discuss the universal bundle gerbe as in [CMi1][CMi2], which provides a

unifying viewpoint for those bundle gerbes constructed from various determinant line bundles.

4. The universal bundle gerbe

Let H be an infinite dimensional separable complex Hilbert space. Let Fa.s∗ be the space of

all self-adjoint Fredholm operators on H with positive and negative essential spectrum. With

the norm topology on Fa.s∗ , Atiyah-Singer [AS1] showed that Fa.s

∗ is a representing space for

the K1-group, that is, for any closed manifold B,

K1(B) ∼= [B,Fa.s∗ ](24)

the homotopy classes of continuous maps from B to Fa.s∗ .

As Fa.s∗ is homotopy equivalent to U (1), the group of unitary automorphisms g : H → H

such that g − 1 is trace class, we obtain

K1(B) ∼= [B,U (1)].(25)

In [CMi1], a universal bundle gerbe was constructed on U (1) with the Dixmier-Douady class

given by the basic 3-form

1

24π2Tr(g−1dg)3,(26)

the generator of H3(U (1), Z).

For any compact Lie group, this basic 3-form gives the so-called basic gerbe. We prefer

however to call it the universal bundle gerbe in the sense that many examples of bundle gerbes

on smooth manifolds are obtained by pulling back this universal bundle gerbe via certain smooth

maps from B to U (1).

Note that the odd Chern character of K1(B) = [B,U (1)] is given by

Ch([g]) =∑

n≥0

[

(−1)n n!

(2πi)n+1(2n + 1)!Tr(

(g−1dg)2n+1)

]

,(27)

for a smooth map g : B → U (1) representing a K1-element [g] in K1(B). This odd Chern

character formula was proved in [Getz].

The following theorem was established in [CMi1] as the first obstruction to obtaining a second

quantization for a smooth family of Dirac operators (parametrized by B) on an odd dimensional

spin manifold. By a second quantization, we mean an irreducible representation of the canonical

anticommutative relations (CAR) algebra, the complex Clifford algebra Cl(H ⊕ H), which is

compatible with the action of the quantized Dirac operator. Such a representation is given by

the Fock space associated to a polarization on H . For a bundle of Hilbert space over B, a

continuous polarization always exists locally. This leads to a bundle gerbe over B.

Theorem 4.1. For any K1-element [g] ∈ K1(B). There exists a canonical isomorphism class

of bundle gerbe whose Dixmier-Douady class is given by the degree 3 part of the odd Chern

character Ch([g]).


Proof. The canonical bundle gerbe we are after is essentially the pull-back bundle gerbe from

the universal bundle gerbe over U (1) under a smooth map g : B → U (1) representing [g].

Consider the polarized Hilbert space H = L2(S1, H) with the polarization ǫ given by the

Hardy decomposition.

H = L2(S1, H) = H+ ⊕H−.

That is we take the polarisation given by splitting into positive and negative Fourier modes.

Then the smooth based loop group ΩU (1) acts naturally on H. From [PS], we see that

ΩU (1) ⊂ Ures(H, ǫ).(28)

Here Ures(H, ǫ) is the restricted unitary group of H with respect to the polarization ǫ, those

g ∈ U(H) such that the off-diagonal block of g is Hilbert-Schmidt. It was shown in [CMi1] that

the inclusion ΩU (1) ⊂ Ures(H, ǫ) is a homotopy equivalence.

We know that the holonomy map from the space of connections on a trivial U (1)-bundle over

S1 provides a model for the universal ΩU (1)-bundle. Hence, U (1) is a classifying space for ΩU (1).

From K1(B) ∼= [B,U (1)], we conclude that elements in K1(B) are in one-to-one correspon-

dence with isomorphism classes of principal Ures(H, ǫ)-bundles over B.

Associated to the basic three form (26) on U (1), is the universal gerbe realized by the central

extension

1 → U(1) → Ures → Ures(H, ǫ) → 1.(29)

Note that the fundamental representation of Ures acts on the Fock space (see [PS])

FH = Λ(H+) ⊗ Λ(H−)

This gives rise to a homomorphism Ures(H, ǫ) → PU(FH) which induces a PU(FH) principal

bundle from the Ures(H, ǫ) principal bundle associated to the K1-element [g].

Denote by Pg the resulting PU(FH) principal bundle corresponding to a smooth map

g : B → U (1) representing [g]. The corresponding lifting bundle gerbe (as defined in [Mur]) is

the canonical bundle gerbe over B for which we are looking.

Recall that the lifting bundle gerbe associated a principal PU(FH)-bundle can be described

locally as follows. Take a local trivialization of Pg with respect to a good covering of B =⋃

α Uα.

Assume over Uα, the trivialization is given by a local section

sα : Uα → Pg|Uα ,

such that the transition function over Uαβ = Uα ∩ Uβ is given by a smooth function

γαβ : Uαβ → PU(FH).

We can define a local Hermitian line bundle Lαβ over Uαβ by the pull-back the complex line

bundle associated to (29) via γαβ . These local Hermitian line bundles Lαβ define a bundle

gerbe over B.

From the odd Chern character formula (27), we know that

Ch([g])(3) =1

24π2Tr(g−1dg)3,

which implies that the Dixmier-Douady class of Pg is exactly the degree 3 part of

Ch([g]) ∈ H3(B, Z).


For an Hermitian vector bundle (V, hV ) over an oriented, closed, spin manifold M with a

Riemannian metric denote by A the space of unitary connections on V and by G the based

gauge transformation group as in section 3.

In [CMi2], an explicit smooth map g : A/G → U (1) was constructed: firstly assign to any

A ∈ A a unitary operator in U (1) by

A → g(A) = −exp(iπ/DA

| /DA| + χ(| /DA|)),

where χ is any positive smooth exponentially decay function on [0,∞) with χ(0) = 1. As g(A)

is not gauge invariant,

g(Au) = u−1g(A)u,

for any u ∈ G, it doesn’t define a smooth map from A/G to U (1). In order to get a gauge invari-

ant map, a global section for the associated U(H)-bundle A×G U(H) is needed. This exists due

to the contractibility of U(H). This section is given by a smooth map r : A → U(H) such that

r(Au) = u−1r(A). Then the required map g : A/G → U (1) is given by g(A) = r(A)−1g(A)r(A).

Let B be a smooth submanifold of A/G. The restriction of g to B defines an element

in K1(B), which is exactly the family of Dirac operators over B associated to the universal

connection A on V (see section 3 for the definition).

From the local family index theorem, we know that

(

∫

M

A(TM,∇T M)Ch(V, A))

(3)=

1

24π2Tr(g−1dg)3

in H3(B, R). One can verify that the canonical bundle gerbe constructed in the proof of The-

orem 4.1 agrees with the determinant bundle gerbe constructed in section 3. In section 2 of

[CMMi1], both of these constructions were presented. Explicit computations of the correspond-

ing Dixmier-Douady class were given in [CMMi1], see [CMi2] for more examples.

5. Index gerbe as induced from the universal bundle gerbe

Let π : X → B be a smooth fibration over a closed smooth manifold B, whose fibers are

diffeomorphic to a compact, oriented, odd dimensional spin manifold M . Let (V, hV ,∇V ) be a

Hermitian vector bundle over X equipped with a unitary connection.

Let gX/B be a metric on the relative tangent bundle T (X/B) and let SX/B be the spinor

bundle associated to (T (X/B), gX/B). Let T HX be a horizontal vector subbundle of TX . Then

(T HX, gX/B) determines a connection ∇X/B on T (X/B) as in section 2.

5.1. Family of Dirac operators on odd dimensional closed manifolds. When the fibers

of π are closed, the family of Dirac operators /Dbb∈B defines a K1-element

Ind( /D) ∈ K1(B),

with Ch(Ind( /D)) = π∗

(


∈ Hodd(B, R) as given by the local

family index theorem (Theorem 2.4).

Then Theorem 4.1 provides a canonical bundle gerbe GM over B whose Dixmier-Douady

class is given by

π∗

(


(3).

One can repeat the construction in section 3 to get the gerbe connection and curving on this

bundle gerbe. This is similar to the construction of Lott in [L].


Now we can prove the following theorem, which was obtained in [L] by Lott using different

construction.

Theorem 5.1. Let π : X → B be a smooth fibration with fibers closed odd dimensional spin

manifolds M with the geometric data given as in the beginning of this section. Then the asso-

ciated family of Dirac operators defines a canonical bundle gerbe GM over B equipped with a

Hermitian metric and a unitary gerbe connection whose curving (up to an exact form) is given

by the locally defined eta form such that its bundle gerbe curvature is given by

(

∫

M


(3).

Proof. Cover B by Uλ (for λ ∈ R) such that

Uλ = b ∈ B| /Db − λ is invertible.

Over Uλ, the bundle of Hilbert spaces of the square integrable sections along the fibers has a

uniform polarization:

Hb = H+b (λ) ⊕ H−

b (λ),

the spectral decomposition with respect to /DMP,b −λ into the positive and negative eigenspaces.

Denote by Pλ the smooth family of orthogonal projections Pλ,bb∈Uλonto H+

b (λ) along the

above uniform polarization over Uλ.

Note that the restriction of Ind( /D) on Uλ is trivial by the result of Melrose-Piazza in [MP1],

as /D over Uλ admits a spectral section. So over Uλ, its Chern character form is exact:

π∗

(


= dηPλeven(30)

where ηPλeven, unique up to an exact form, is the even eta form on Uλ, associated to /DM

b b∈Uλ

and the spectral section Pλ.

For λ ≥ λ′, over Uλλ′ = Uλ∩Uλ′ , consider the fibration [0, 1]×X → B with even dimensional

fibers, the boundary fibration has two components 0×X and 1×X . The spectral section

can be chosen to be

Pλλ′ = Pλ ⊕ (I − Pλ′).(31)

Associated with this spectral section Pλλ′ , there exists an even eta form ηPλλ′

even on Uλλ′ ,

modulo exact forms, depending only on Pλλ′ and the family of boundary Dirac operators.

From the definition of the eta form, we have

ηPλλ′

even = ηPλeven − ηPλ′

even,

which should be understood modulo exact even forms on Uαβ .

Then the family of Dirac operators with boundary condition given by the spectral section

Pλλ′ , denoted by

/D[0,1]×MPλλ′ ,b b∈Uλλ′ ,

is a smooth family of Fredholm operators, which has a well-defined index Ind( /D[0,1]×MPλλ′

) in

K0(Uλλ′ ).


The corresponding determinant line bundle over Uλλ′ , denoted by Detλλ′ , is a Hermitian

line bundle equipped with the Quillen metric and the Bismut-Freed unitary connection. Its

first Chern class is given by the local family index formula (18):[

π∗

(

A(T ([0, 1]× X/Uλλ′),∇X/B)Ch(V,∇V ))

− ηPλλ′

even

]

(2).(32)

Note that in this situation, the contribution from the characteristic class

π∗

(

A(T ([0, 1]× X/Uλλ′),∇X/B)Ch(V,∇V ))

vanishes, as it has no component in [0, 1]-direction. Hence, the first Chern class of Detλλ′ is

represented by the(

ηPλλ′

even

)

(2)=(

ηPλ′

even − ηPλeven

)

(2).

We understand that these eta forms are well-defined only modulo exact 2-forms, so the above

equation should be viewed at the cohomological level. This implies that the even eta form, up

to an exact 2-form,(

ηPλeven

)

)(2),

which is only defined over Uλ, is the curving (up to an exact 2-form) for the Bismut-Freed

connection on Detλλ′ . Hence, the gerbe curvature is given by

d(ηPλeven)(2) = d(ηPλ′

even)(2) =(

∫

M


(3).

5.2. Family of Dirac operators on odd dimensional manifolds with boundary. Now

we assume that the fibers of the fibration π : X → B are odd dimensional manifolds with

non-empty boundary.

A Cl(1)-spectral section P for the family of boundary Dirac operators /D∂Mb b∈B provides

a well-defined index for the family of self-adjoint Fredholm operators /DMP :

Ind( /DMP ) ∈ K1(B).

Then Theorem 4.1 provides a canonical bundle gerbe GMP over B whose Dixmier-Douady

class is given by the local family index theorem 2.7:(

π∗(A(T (X/B),∇X/B)Ch(V,∇V )) − ηPodd

)

(3).(33)

Here we should emphasize that ηPodd is an odd eta form associated to a perturbation of

/D∂Mb b∈B by a family of self-adjoint smooth operators AP,bb∈B such that /D∂M

b + AP,b

is invertible as in [MP2]. Note that previously, we always define ηPodd up to an exact form.

It would be interesting to understand the bundle gerbe connection and its curving such that

the bundle gerbe curvature is given by (33). Note that the argument for the previous case can’t

be applied here, as now [0, 1] × M is a manifold with corner near ∂M , the local family index

theorem for such a manifold is not understood, and the even eta form for a family of Dirac

operators on odd dimensional manifolds with boundary which transgresses the odd local index

form (33) has not been found.

For a single manifold with corners, Fredholm perturbations of Dirac operators and their

index formulae have been developed by Loya-Melrose [LM]. For a family of Dirac operators

on a manifold with corners, even in the case of corners up to codimension two, their Fredholm

perturbations and the corresponding local family index theorems are still open.


Instead, we will apply the bundle gerbe theory to find this even eta form which transgresses

the odd local index form (33) and discuss its implications for local family index theory for a

family of Fredholm operators on a manifold with corners of a particular type, [0, 1]× M .

For the fibration π : X → B with odd dimensional manifolds with non-empty boundary as

fiber, the Melrose-Piazza Cl(1)-spectral section P for the family of boundary Dirac operators

/D∂Mb b∈B gives rise to the family of self-adjoint Fredholm operators /DM

P,bb∈B with discrete

spectrum.

Cover B by Uλ with λ ∈ R such that

Uλ = b ∈ B| /DMP,b − λ is invertible.

Note that, over Uλ, the family of self-adjoint Fredholm operators /DMP,b has trivial index in

K1(Uλ). This follows from the fact that, over Uλ, we have a uniform polarization of Hilbert

space of the square integrable sections along the fibers with boundary condition given by the

Cl(1)-spectral section P ,

HP,b = H+P,b(λ) ⊕ H−

P,b(λ),

the spectral decomposition with respect to /DMP,b −λ into the positive and negative eigenspaces.

Denote by Pλ the smooth family of orthogonal projections Pλ,bb∈Uλonto H+

P,b(λ) along the

above uniform polarization over Uλ.

Hence, there exists an even form ηP,λeven on Uλ, unique up to an exact form, satisfying the

following transgression formula over Uλ:

dηP,λeven = π∗(A(T (X/B),∇X/B)Ch(V,∇V )) − ηP

odd.(34)

We remark that these ηP,λ should be even eta forms a la Melrose-Piazza’s construction

in ([MP2]) associated with the family of self-adjoint Fredholm operators /DMP,bb∈Uλ

and the

spectral section Pλ over Uλ.

From the abstract bundle gerbe theory in [Mur], we know that this locally defined even

form(

ηP,λeven

)

(2)on Uλ is the curving, up to an exact form on Uλ, for the bundle gerbe induced

from the universal gerbe as in Theorem 4.1 equipped with a gerbe connection, such that gerbe

curvature is given by

d(ηP,λeven)(2) =

(

∫

M

A(T (X/B),∇X/B)Ch(V,∇V ) − ηPodd

)

(3).

Consider the fibration [0, 1] × X → B with even dimensional fibers. Now the fibers are

manifolds with corners [0, 1] × M , with two codimension two corners given by

0 × ∂M, 1 × ∂M,

and three codimension one faces

0 × M, 1 × M and [0, 1]× ∂M.

For λ ≥ λ′, over Uλλ′ , associated with the codimension one boundary fibration corresponding

to components 0×X and 1×X , we can choose a spectral section for the codimension one

boundary Dirac operators /D0×MP ⊕ /D

1×MP to be

Pλλ′ = Pλ ⊕ (I − Pλ′).(35)

The corresponding family of Dirac operators

/D[0,1]×M,+P,Pλλ′;b

b∈Uλλ′ ,


over Uλλ′ can be written as a family of operators of the form

∂

∂t+ /DM

P,b : L21([0, 1] × Xb, (S

+ ⊗ V )|Xb; P, Pλλ′ ) → L2([0, 1]× Xb, (S

− ⊗ V )|Xb)(36)

where L21([0, 1]×Xb, (S

+ ⊗V )|Xb; Pλλ′) denotes the L2

1-integrable sections with boundary con-

ditions given by P along ∂(Xb) and Pλλ′ along 0 × Xb ⊔ 1 × Xb.

Note that /DMP,bb∈B is a family of self-adjoint pseudodifferential Fredholm operators with

discrete spectrum. From the explicit time evolution equation associated with∂

∂t+ /DM

P,b, we

can see that /D[0,1]×M,+P,Pλλ′;b

b∈Uλλ′ is a family of Fredholm operators, which induce a determinant

line bundle from the universal determinant line bundle over the space of Fredholm operators,

denoted by DetPλλ′ .

It would be interesting to know whether the methods of this paper can be extended to answer

the following:

(a) Is there a Quillen-type metric and its unitary Bismut-Freed-type connection on DetPλλ′?

(b) Is the curvature of this unitary Bismut-Freed-type connection given by a yet to be

defined local family index theorem for manifolds with corners [0, 1] × M such that the

degree 3 part implies the transgression formula (34)?

Both these questions are also very important from the index theory viewpoint (see for ex-

ample [LM] [Bu2]).

Remark 5.2. From Proposition 4 in [MP2], we know that if P1 and P2 are two Cl(1)-spectral

sections for /D∂Mb b∈B, then the Atiyah-Singer suspension operation [AS1] defines a difference

element

[P2 − P1] ∈ K1(B),

such that

Ind( /DMP1

) − Ind( /DMP2

) = [P2 − P1] ∈ K1(B),

whose Chern character is given by ηP2

odd − ηP1

odd. It was also shown in Proposition 12 in [MP2]

that for a fixed P1, as P2 ranges over all Cl(1)-spectral sections, [P2 − P1] exhausts K1(B).

Apply Theorem 4.1 again to [P2 − P1], we obtain a canonical bundle gerbe GP2P1associated

to [P2 − P1] ∈ K1(B) such that its Dixmier-Douady class is given by(

ηP2

odd − ηP1

odd

)

(3).

Moreover, we have the following isomorphism relating the bundle gerbes associated to families

of Dirac operators on odd-dimensional manifolds with boundary equipped with two different

Cl(1)-spectral sections for the family of boundary Dirac operators

GMP1

∼= GMP2

⊗ GP2P1.

Take a smooth fibration X over B, with closed odd-dimensional fibers partitioned into two

codimensional 0 submanifolds M = M0∪M1 along a codimension 1 submanifold ∂M0 = −∂M1.

Assume that the metric gX/B is of product type near the collar neighborhood of the separating

submanifold. Let P be a Cl(1)-spectral section for /D∂M0

b b∈B, then I − P is a Cl(1)-spectral

section for /D∂M1

b b∈B. Then we have the following splitting formula for the canonical bundle

gerbe GM obtained in Theorem 5.1:

GM ∼= GM0

P ⊗ GM1

I−P .


Compare this with the corresponding splitting formula (19) for the determinant line bundle for

the splitting of a smooth fibration with closed even-dimensional fibers.

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A.L. Carey,

Mathematical Sciences Institute, Australian National University, Canberra ACT, Australia

[email protected]

B.L. Wang,

Institut fur Mathematik, Universitat Zurich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland

[email protected]

carey wang

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