equivariant localization in supersymmetric quantum mechanics1228581/fulltext01.pdf · equivariant...

Equivariant Localization in SupersymmetricQuantum Mechanics

Bachelor Thesis in Physics, 15c

Emil HössjerDepartment of Physics and Astronomy

Division of Theoretical Physics

Supervisor: Konstantina Polydorou

Subject reader: Marco Chiodaroli

June 28, 2018

Abstract

We review equivariant localization and through the Feynman formalism of quantum me-chanics motivate its role as a tool for calculating partition functions. We also considera specific supersymmetric theory of one boson and two fermions and conclude that byapplying localization to its partition function we may arrive at a known result that haspreviously been derived using different approaches. This paper follows a similar articleby Levent Akant [1].

Sammanfattning

Vi studerar ekvivariant lokalisering och motiverar medelst Feynmans vägintegralsforma-lism av kvantmekanik lokaliseringens användning till att beräkna partitionsfunktioner. Vibetraktar därefter ett särskilt supersymmetriskt system beståendes av en boson och tvåfermioner och tillämpar lokalisering för att beräkna partitionsfunktionen. Detta leder tillett resultat som tidigare har visats med hjälp av andra metoder. Uppsatsen är inspireradav en artikel av Levent Akant [1].

Contents

1 Introduction 1

2 Mathematical Preliminaries 22.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Actions on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Infinitesimal actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Grassmann algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Equivariant Cohomology 133.1 Cartan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Equivariant symplectic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Equivariant Localization 19

5 Equivariant Cohomology in Quantum Mechanics 255.1 Path integral formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Loop space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Equivariant Localization in Supersymmetric Quantum Mechanics 29

7 Conclusion 32

1 INTRODUCTION

1 Introduction

We start this paper with a math problem:ż 1

´1x cosxdx.

Although it is not that much of a dilemma to find the primitive function, evaluating it at theboundary to find that the contributions cancel would annoy any mathematician, as one hadthen missed an ’obvious’ symmetry in the problem. There is an unofficial ruling in math thatif a problem possesses symmetry, one should exploit this to simplify the problem. This paperis all about that, in the context of equivariant cohomology.

The symmetries we are considering are manifolds that permit Lie group actions, and theassociated problems are integrals over the manifolds. We want to distinguish those differentialforms for which the problem of integration may be simplified - this will be the cohomology -and see what this simplification means - this will be the localization. Denoting the manifoldand the Lie group M and G respectively, it is a known result that if G acts freely on M thenthe quotient MG is a smooth manifold. In this case the new, equivariant, cohomology thatwe want to define is

H˚eqpMq :“ H˚pMGq,

i.e. the ordinary de Rham cohomology of the quotient space. If the action is not free howeverthis definition fails, and a more refined one is needed. But the above equation is still themotivational one that in the case of free action should be valid.

The first equivariant localization formulae came in the early 1980’s and were due to Duister-maat and Heckman on one side, Berline, Vergne; Atiyah, Bott on another. These powerfultheorems show how certain integrals, largely motivated by physics, may be reduced to dis-crete sums. Equivariant cohomology has since been an active field of research attended bymathematicians and physicists alike.

According to the Feynman formalism of quantum mechanics, transition amplitudes may becalculated through an infinite dimensional integral, called a path integral. Not surprisingly,this path integral is very difficult to solve - at the time it was introduced, the only knownexamples were the harmonic oscillator and the free particle! What one wants to calculate inquantum physics is generally the partition function. Expressed in the Feynman formalism thepartition function too becomes an infinite dimensional integral. Using a localization theoremto, if possible, simplify such an integral is therefore very appealing.

We start this paper by reviewing some of the mathematics required for equivariant cohomology.We assume the reader knows ordinary de Rham theory and Lie group basics. We do howevercover some examples of Lie groups that we will carry with us to later chapters and we coverLie group actions on manifolds. The chapter on symplectic geometry is what links physicsto geometry and enables the application of cohomological results to physics. The chapter onGrassmannians is important both to the mathematics and the physics of this paper.

We use the Cartan model for equivariant cohomology and discuss some of its features, espe-cially integration. We also consider equivariant cohomology for symplectic manifolds. Regard-ing the localization we prove the BVAB theorem already mentioned (Berline, Vergne; Atiyah,Bott).

1

2 MATHEMATICAL PRELIMINARIES

We then get into some physics. We motivate why equivariant cohomology in general might berelevant to calculating partition functions, and we see how a specific supersymmetry by itselfseems to generate a setting of equivariant cohomology. In the final chapter we take a commonexample of a supersymmetric system and combine the two ideas above to put the partitionfunction into a setting of equivariant cohomology. Finally, we apply the BVAB theorem toshow that the partition function localizes.

2 Mathematical Preliminaries

2.1 Lie groups

Most Lie groups of interest are matrix Lie groups, where the operations are simply matrixmultiplication and matrix inverse, and the topological structure as a manifold comes fromthe identification of a matrix with an element of R2n or C2n. Examples include the group ofinvertible nˆn- matrices GLpnq and some of its subsets the orthogonal group Opnq, which is areal matrix Lie group, and the unitary group Upnq, which is a complex one. We also have themore intricate matrix Lie group Op1, 3q of transformations on R4 preserving the Minkowskimetric and SppV q, the set of symplectomorphisms from a vector space V to itself, which is amatrix lie group under a fixed basis.

Let us review some examples more thoroughly, together with some of the key concepts.Example 2.1. The circle group S1 is the circle in the complex plane

S1 :“ tz P C : |z|2 “ 1u

with group multiplication inherited as the usual multiplication from the complex numbers,well defined as

|z1 ¨ z2|2 “ |z1|

2|z2|2 “ 1.

Note that we may also writeS1 “ teit P C : t P r0, 2πqu.

Example 2.2. The group SOp2q is the matrix Lie group of orthogonal matrices with positivedeterminant, that is, linear maps on R2 preserving both norm and orientation; known to bethe set of rotations in the plane!Example 2.3. Consider R with the operation of regular addition, `. This is a familiar groupwhose operations

a ¨ b :“ a` b

anda´1 :“ ´a

are commonly known smooth functions of calculus. Thus G “ R with this operation is a Liegroup. The identity for G is e “ 0 and the Lie algebra is g “ T0G ” R. Let us find theexponential map for some X P g. In local coordinate we have X “ s ddx |0 for some scalar s.Let c be the curve cptq “ ts in G so cp0q “ 0 and c1p0q “ X and calculate the left invariantvector field X at a point a P G by

X|a “ La˚X “d

dtpLa ˝ cq|t“0 “

d

dtpa` tsq|t“0 “ s

d

dx|a.

2

2 MATHEMATICAL PRELIMINARIES 2.1 Lie groups

Thus X is constant wrt the standard coordinate on R,

X “ sd

dx,

and the exponential map satisfies

d

dtexpptXq|t0 “ X|exppt0Xq “ s

d

dx|exppt0Xq.

Together with the condition expp0q “ 0 we see that we must have expptXq “ ts.

2.1.1 Actions on manifolds

Definition 2.4. A Lie group G can, beyond act on itself, act on another manifold M . Wecall this the action of G on M and write

GˆM ÑM

g ˆ pÑ g ¨ p,

if this map is smooth and satisfies the two criteria

e ¨ p “ p (1)

pghq ¨ p “ g ¨ ph ¨ pq, (2)

for g, h P G.Definition 2.5. We say that the action of G on M is free if for all p PM

g ¨ p ùñ g “ e,

i.e. no point is non-trivially fixed under action of G.Example 2.6. As hinted about in example (2.2), the Lie group SOp2q of rotations may acton the manifold R2 by matrix multiplication. Clearly this satisifies criteria (1) and (2):

ˆ

1 00 1

˙ˆ

ab

˙

“

ˆ

ab

˙

and

pABq

ˆ

ab

˙

“ A

ˆ

B

ˆ

ab

˙˙

.

We know from intuition that this action has one fixed point, at origo. Thus this action is notfree.Example 2.7. The circle group S1 from example (2.1) may act on the complex plane C bymultiplication:

eit ¨ z “ eitz “ eitrpcos θ ` i sin θq “ rpcospθ ` tq ` i sinpθ ` tqq.

3


Note that both S1 and SOp2q act by rotation. If we identify C ” R2 in the standard wayas real manifolds, we see that these Lie groups act in exactly the same way. We also see,since

S1 “ teit P C : t P r0, 2πqu,

that we may identifyS1 ” R mod p2πq

where the group action on pR mod p2πqq is additive and modulo 2π:

t1 ¨ t2 “ t1 ` t2 mod p2πq.

This is an important identification that we will make much use of.Example 2.8. Using the above identification S1 ” R mod p2πq, we may use example (2.3)to obtain the exponential map on S1. For any X P TeS

1 ” R with X “ s ddx |e:

expptXq “ ts mod p2πq.

The Lie algebra induces a flow and a vector field on the manifold in much the same way as itdoes on the Lie group.Definition 2.9. Let G act on M and X P g. Then the exponential map at p PM induced byg is

pptq “ exp pXtq ¨ p

and the corresponding induced vector field is

X#ptq “d

dt

`

exp pXtq ¨ p˘

|t“0.

Since the exponential map on G is a flow this makes the corresponding exponential map onM a flow as well. This in turn ensures that the induced vector field is well defined.Example 2.10. Let us consider the action of S1 on the sphere S2 by rotation through thez-axis. The action is free except at the north and south pole of the sphere. We use

S1 ” R mod p2πq.

In spherical coordinates, a point p P S2 has

p “ pφ, θq

and t P S1 acts byt ¨ p “ pφ, θ ` tq mod p2πq.

Take as before X “ s ddx |e P TeS1. Using the exponential map on S1 that we obtained in

example (2.8) we get the induced flow

σpt, pq “ expptXq ¨ p “ ts ¨ p “ pφ, θ ` tsq mod p2πq

and the induced vector field

X#|p “d

dtσpt, pq|t“0 “ p0, sq “ s

B

Bθ|p.

4


The action of a group on a manifold as in definition (2.4) induces an action of the groupon certain related structures, specifically the Lie algebra, the Lie coalgebra and the exterioralgebra. We will study these here.

The Lie algebra. Consider the conjugation map κg0 which for fixed g0 P G maps g P G toG by

κg0g Ñ g0gg´10 .

Observe that κg0e “ e so the pushforward κg0˚ at e is a map

κg0˚|e : TeGÑ TeG

and identifying g “ TeG we have, for every g P G, obtained an induced map κg˚ on g. Thiswill be called the adjoint action of G on g, denoted Adg :“ κg˚, and for X P g

g ¨X “ AdgX.

It is straight forward to check the criteria in definition (2.4) for this group action. Wehave

e ¨X “ AdeX “d

dtpκeexpptXqq|t“0 “

d

dtpe expptXqe´1q|t“0 “ X

which shows (1) andκghp “ ghppghq´1 “ gphph´1qg´1 “ kgkhp

together with the chain rule gives

Adgh “ κgh˚ “ κg˚κh˚ “ AdgAdh

which shows (2).

The Lie coalgebra. The adjoint action directly gives a corresponding coadjoint action of Gon the dual space of the lie algebra g˚ by, for X P g, φ P g˚, requiring

pg ¨ φqpXq “ φpg´1 ¨Xq,

or denoting the coadjoint action Ad#g ,

pAd#g φqpXq “ φpAdg´1Xq. (3)

This is easily seen to be well defined by picking a basis pXiqdimpGqi“1 in g and a basis dual to

it pφiqdimpGqi“1 P g˚ and checking that Ad#g φ exists and is uniquely defined. Since the adjoint

action is a pushforward it is linear and it is enough to consider the case X “ Xa, φ “ φb oftwo basis elements. We have

Adg´1Xa “ caiXi (4)

for some constants cai and so the righthandside of (3) is

φbpAdg´1Xaq “ cab .

This equation now readspAd#

g φbqpXaq “ cab

5


so we must haveAd#

g φb “ cibφ

i

with the same constants cji as in (4) which uniquely determines the coadjoint action and sofinishes the claim. We also see that the adjoint and the coadjoint actions correspond to thesame linear transformation in the respective dual basis.

The first criterion (1) in definition (2.4) follows immediately from (3) as

pAd#e φqpXq “ φpAde´1Xq “ φpXq ùñ Ad#

e φ “ φ

and the second criterion (2) follows from

Ad#ghφpXq “ φpAd

pghq´1Xq,

which from the corresponding property of the adjoint map just showed equals

φpAdpghq´1Xq “ φpAdh´1Adg´1Xq “ Ad#

h φpAdg´1Xq “ Ad#g Ad

#h φpXq

soAd#

gh “ Ad#g Ad

#h .

The exterior algebra. An element g P G acts on a form ω P Ω˚pMq by pullback of theinverse element:

g ¨ ω “ g´1˚ω.

Criterion (1) is obviously satisfied and (2) follows from

gh ¨ ω “ pghq´1˚ω “ ph´1g´1q˚ω “ g´1˚h´1˚ω “ g ¨ ph ¨ ωq.

Remark: If the group action is commutative then clearly the conjugation map κg is just theidentity, and the same follows for the adjoint and the coadjoint actions.Example 2.11. For a matrix lie group G acting by matrix multiplication we have for g P G,X P g

AdgX “d

dtpκgexpptXqq|t“0 “

d

dtpg expptXqg´1q|t“0 “

d

dtpexpptgXg´1qq|t“0 “ gXg´1.

where in the third equality we expand the exponential in a sum and use gXng´1 “ pgXg´1qn.Example 2.12. Let G act on M and take f P Ω0pMq a function. For g P G and p P M wehave one action fpg ¨ pq and another g ¨ fppq “ g´1˚fppq “ fpg´1 ¨ pq, related by

fpg ¨ pq “`

g´1 ¨ f˘

ppq.

2.1.2 Infinitesimal actions

Assume G acts on a manifold M and let g “ exppXq for some X P g. Consider the action ofgptq :“ expptXq on a point p PM :

gptq ¨ p “ expptXq ¨ p. (5)

6


Taking the derivative at t “ 0 we obtain

d

dtgptq ¨ p|t“0 “ X#|p, (6)

with X# the corresponding fundamental vector field to X at p. So a first order approximationof the action of g to p is

gptq ¨ p « p` tX#,

which is accurate for an infinitesimal value on t. We say that X is the infinitesimal generatorof the action g, referring to (5), and the infinitesimal action of g on the manifold M is X#,referring to (6). Motivated by this, we generalize as follows.Definition 2.13. Let g P G act on x P N . If g is in the image of the exponential map,g “ exppV q, we define the infinitesimal action of g on N as

d

dtpexpptV q ¨ xq|t“0.

Example 2.14. Let us consider the infinitesimal action of g “ expptV q on a form ω in theexterior algebra. We have

d

dtpexpptV q ¨ ωq|t“0 “ lim

tÑ0

expp´tV q˚ω ´ ω

t,

which is the definition of the lie derivative of ω along the induced vector field V #, £V #ω.Definition 2.15. We say that a point x P N is invariant under group action g P G if g ¨x “ x.We also say that x is infinitesimally invariant under V P g if the infinitesimal action on x is0.

It follows immediately from the definition that finite group action invariance implies infinites-imal invariance. We will now show an important converse.Theorem 2.16. Consider the image of the exponential map H :“ Impexpq Ď G. If points inN are invariant under infinitesimal action, then they are invariant under group actions in H.

Proof. For any x P N and any V P g we have

d

dtpexpptV q ¨ xq|t“0 “ 0. (7)

We want to show that g ¨ x “ x for g P H. By definition of H, g “ exppV q for some V in theLie algebra. Consider the map gptq “ expptV q. If

d

dtpgptq ¨ xq “ 0

for all 0 ď t ď 1 then we are done as we must have gp1q ¨ x “ x. Let us therefore showthat

d

dtpgptq ¨ xq|t“t0 “ 0

for an arbitrary t0. We have

d

dtpgptq ¨ xq|t“t0 “

d

dtpgpt` t0q ¨ xq|t“0 “

d

dtpexpppt` t0qV q ¨ xq|t“0

“d

dtpexpptV q ¨ pexppt0V q ¨ xqq|t“0

“d

dtpexpptV q ¨ x1q|t“0 “ 0,

7

2 MATHEMATICAL PRELIMINARIES 2.2 Symplectic geometry

where in the last line we put x1 “ exppt0V q ¨ x, and the whole thing vanishes due to (7).

Corollary 2.16.1. For any compact and connected Lie group G, invariance and infinitesimalinvariance of the action are equivalent.

Proof. Follows since the exponential map on any both compact and connected manifold issurjective and theorem (2.16) just proved. In the context of this theorem, H “ G.

2.2 Symplectic geometry

Consider the following 2-form on R2n with coordinates pqi, piqni“1:

ω “ dqi ^ dpi (8)

whose matrix elements are constant

pωijq “

ˆ

0 ´EnEn 0

˙

.

This form is called the standard symplectic form on Euclidean space and enjoys several usefulproperties

1q ωij “ ´ωji, pantisymmetryq

2q detωij ‰ 0, pnondegeneracyq

3q dω “ 0, pclosednessq

The usefulness claim is legitimized by classical mechanics where the problem of finding theequations of motion of a specific system is made into a geometrical one, in a setting calledsymplectic geometry. Namely, the dynamics of an n-coordinate system pq1, . . . , qnq may besolved from the Hamiltonian H in phase space M “ pq1, . . . , qn, p1, . . . pnq, by solving thedifferential equations

BH

Bpi“ 9qi (9)

´BH

Bqi“ 9pi (10)

This may be expressed using the symplectic form ω above and Poisson brackets t¨, ¨uω definedby

tf, guω “ pωijq´1 Bf

BqiBg

Bpj

as for any function on phase space f P Ω0pMq we then have

9f “ tf,Huω.

Specifically the cases f “ qi and f “ pi yield the Hamilton equations (9)-(10). Beforediscussing this any further, Let us define a symplectic form generally.

8


Definition 2.17. A symplectic form on a 2n-dimensional manifold M is a nondegenerate2-form ω that is closed

dω “ 0.

A manifold M that admits a symplectic form is called a symplectic manifold.

It turns out that a symplectic manifold always describes a mechanical system, as by thefollowing theorem it may be put in the standard form (8) and be interpreted as a phase space[6].Theorem 2.18 (Darboux Theorem). For a symplectic manifold pM,ωq we may locally findcoordinates pqi, piqni“1 so that

ω “ dqi ^ dpi.

This geometric approach to mechanics will now grant the following two advantages. Wemay independently of coordinates state the dynamics of a mechanical system pM,ωq withHamiltonian H as

9f “ tf,Huω

for an observable f of the system. And the shift from the classical formalism to a quantumformalism may now be made. This process is called geometric quantization and although wewill not go into details results in the following simple transition. Functions become operatorsand the Poisson bracket becomes a Lie bracket:

f Ñ f

9f “ i~rf , Hsω.

This result will be used in chapter 5 when we go into quantum dynamics. We end this chapterwith a couple more useful concepts stemming from symplectic manifolds and an example fromgeometry.

One can easily show that an even dimensional nondegenerate form is not nilpotent unless theincurred degree is above that of the space, in which case it is definitely zero. Specifically onecan show for a symplectic manifold pM,ωq of dimension 2n that

ωn “ cpxqdx1 ^ . . .^ dx2n ‰ 0

is a top form with coefficient function c [6]. Passing to Darboux coordinates we obtain

ωn “ pdqi ^ dpiqn “ n! dq1 . . . dqndp1 . . . dpn

so a symplectic manifold has a natural volume form called the Liouville volume form µLdefined by

µL “1

n!ωn. (11)

In arbitrary coordinates this becomes

µL “b

detωijpxqdx1 . . . dx2n. (12)

9


Example 2.19. Let us show that the sphere S2 is symplectic. The volume form on R3 inducesan area form on S2 through

ω :“ ιNdx^ dy ^ dz

where N is the normal unit vector to the sphere in the outer direction. Thus N “ px, y, zq atpx, y, zq and

ω “ xdy ^ dz ´ ydx^ dz ` zdx^ dy.

Passing to local spherical coordinates φ, θ with for example

x “ sinφ cos θ

dx “ cosφ cos θdφ´ sinφ sin θdθ

and substituting into ω we obtain

ω “ sinφdφ^ dθ. (13)

Clearly this satisfies being a symplectic form and the Liouville form µL “ ω1 is the same asthe euclidean area form as

ż

S2

ω “

ż 2π

θ“0

ż π

φ“0sinφdφ^ dθ “ 4π.

Observe that the Darboux coordinates should here satisfy

dq “ sinφdφ

dp “ dθ

and thus

q “ cosφ

p “ θ

with the standard symplectic form

ω “ dq ^ dp “ d cosφ^ dθ.

Definition 2.20. Let pM,ωq be symplectic with a group action G that preserves ω, i.e.

g ¨ ω “ ω ðñ g´1˚ω “ ω.

Also let µ : M Ñ g˚ and define for X P g the function µX P Ω0pMq by

µX |p :“ ιXµ|p.

For X# the induced vector field on M , µ is called a moment map if

1q dµX “ ιX#ω (14)2q µ|g¨p “ g ¨ µ|p. (15)

Criterion (2) is that µ should be an equivariant map and is equivalent to

µ|gp “ Ad#g µ|p.

10

2 MATHEMATICAL PRELIMINARIES 2.3 Grassmann algebra

Example 2.21. Let us include a group action of S1 in example (2.19) just studied to obtain amoment map on the symplectic manifold S2. This same group action was studied in example(2.8) where we found that the induced vector field by X “ s ddx |e P s

1 ” R in S2 is

X# “ sB

Bθ.

Thus taking the symplectic form in eq. (13) gives for the moment map

dµX “ ιX#ω “ sinφdφ^ dθpsB

Bθq “ ´s sinφdφ.

Integrating yieldsµX “ s cosφ` sλ,

where we include the s in the constant since the moment map at any point p P S2 should belinear as a map from the lie algebra s1. We may then conclude

µ “ cosφ` λ P R˚

where we identify s1˚ “ R˚.Remark. If the Lie algebra is finitely generated by some Xa then it is enough to consider themoment map evaluated at these, as for X “ caX

a we have

µX “ caµXa

following the linearity of the dual Lie algebra.

2.3 Grassmann algebra

In our further discussion we are going to need a new type of numbers, anticommuting onesgenerally called Grassmann numbers. In physical contexts these numbers can be interpretedas fermions, whereas usual commuting numbers in C are interpreted as bosons. This is dueto the Pauli principle. We will define the basics of these numbers and state the importanttheorems that will be of use later. For proofs we refer to Nakahara [3].Definition 2.22. A Grassmann algebra Λ is an algebra over some field generated by finitelymany numbers pη1, . . . , ηnq satisfying

ηiηj “ ´ηjηi.

The generators ηi are called Grassmannians.

The prominent and intuitive example of a Grassmann algebra is the exterior algebra overa manifold whose generators can be taken as linearly independent 1-forms, e.g. the dxi.Derivation and integration in a Grassmann algebra are funnily enough defined to be equivalent.We have

ż

dηiηj “BηjBηi

:“ δij (16)ż

dηi1 “1

Bηi:“ 0. (17)

11

2 MATHEMATICAL PRELIMINARIES 2.3 Grassmann algebra

So derivation and integration of scalars vanish and the volume of a Grassmann algebra isnormalized:

ż

Λdηn . . . dη1η1 . . . ηn “ 1.

From this we have the following important properties of Grassmann integration.Proposition 2.23. A linear change of coordinates ηi “ aijη

j satisfies

dnη “1

det aijdnη. (18)

A Gaussian integral in Grassmannians satisfiesż

Λd2nηe

12aijη

iηj “a

det aij . (19)

These should be compared with their commutating number analogues

dnx “ det aijdnx (20)

andż

dnxe´aijxixj “

πn2a

det aij. (21)

We will now show a useful way of rewriting an ordinary integral. A k-form α P ΩkpMq maybe written

α “ fi1...ikpxqdxi1 . . . dxik

where as noted before the dxi are part of a Grassmann algebra. We may therefore let ηi “ dxi

and consider α as an element of Ω0pMq b Λ,

αpx, ηq “ fi1...ikpxqηi1 . . . ηik . (22)

We may then use V olpΛq “ 1 to rewrite an integral over M as an integral over M ˆ Λ,according to the following proposition.Proposition 2.24. For an integrable form α on M and the identification in eq. (22) we have

ż

Mα “

ż

MˆΛdnxdnηαpx, ηq.

Proof. Assume first degpαq ă n “ dimpMq. Then the left side of the claimed equalityvanishes, and after integrating out the existing grassmann numbers in αpx, ηq we are left witha scalar that vanishes according to eq. (17).Assume then degpαq “ n. Thus α “ fpxqdx1 . . . dxn and the right side of the propositionreads

ż

MˆΛdnxdnηfpxqη1 . . . ηn “

ż

Mdnxfpxq “

ż

Mα

as the Grassmannians can just be integrated out to yield unity.

The space M ˆ Λ with both commuting and anticommuting coordinates is a supermanifoldand will be called the odd tangent bundle over M , denoted ΠTM .

12

3 EQUIVARIANT COHOMOLOGY

3 Equivariant Cohomology

Let, as when we were discussing Lie groups, G act on M . We want to introduce a cohomologyon M that in the case of free action is

H˚eqpMq “ H˚pMGq. (23)

When the action is not free there is an intuitive way of patching the above definition. Take acontractible space E on which G does act freely, and consider the space

M ˆ E.

The Lie group G acts on each factor independently in this space, called a diagonal action, andis thus free, and since M ˆ E is homotopy equivalent to M we have

H˚pMq ” H˚pM ˆ Eq.

Hence it becomes natural to define

H˚eqpMq “ H˚ppM ˆ EqGq, (24)

which clearly is the same as (23) in the case of free action. This definition can both be shownto be independent of the choice of contractible space E, and to be always applicable, i.e. sucha space E can always be found [2]. This definition although elegant does not open itself to easycalculations. Instead we will go on to look at a model called the Cartan model, which usesa differential complex and a new differential operator that in the same way as the de Rhamcomplex calculates the cohomology. According to a theorem of Cartan these cohomologies arethe same.

3.1 Cartan model

Definition 3.1. Let M and N be two manifolds on which the Lie group G has an action. Amap between these manifolds, f : M Ñ N is said to be equivariant with respect to G if itcommutes with the group action, i.e.

fpg ¨ pq “ g ¨ fppq. (25)

Definition 3.2. An equivariant differential form is a map α : gÑ Ω˚pMq that is equivariant,i.e. for X P g

αpg ¨Xq “ g ¨ αpXq ðñ αpAdgXq “ g´1˚αpXq. (26)

The set of such forms is called the equivariant differential complex and is denoted Ω8G pMq.

We will show that the equivariant differential complex, like the ordinary de Rham complex,constitutes an algebra under the wedge product and that we may introduce a differentialoperator on it, justifying the name of a differential complex. We will then see how the veryexplicit subset - in fact subalgebra - of polynomial maps of Ω8G pMq has the correct cohomologyin accordance with (23) and (24).

13

3 EQUIVARIANT COHOMOLOGY 3.1 Cartan model

Definition 3.3. We extend the wedge product to equivariant forms by, for α, β P Ω8G pMq

`

α^ β˘

pXq :“ αpXq ^ βpXq.

This is well defined as a direct computation shows that the product is too an equivariantform:

`

α^ β˘

pg ¨Xq “ αpg ¨Xq ^ βpg ¨Xq

“ g ¨ αpXq ^ g ¨ βpXq “ g ¨`

pα^ βqpXq˘

.

Definition 3.4. We define the equivariant differential operator dG on Ω8G pMq by, for α anequivariant form,

pdGαqpXq “ pd` ιX#qpαpXqq.

We must similarly show that this is well defined by showing that the equivariant complex isclosed under dG. Direct computation gives

pdGαqpg ¨Xq “ pd` ιpAdgXq#qpαpg ¨Xqq “ pd` ιpAdgXq#qpg ¨ αpXqq,

which we would like to be equal to

g ¨“

pd` ιX#qpαpXqq‰

.

Now, the ordinary differential commutes with pullpacks which solves the d-term. For theinterior product-term we have the operator identity

ιpAdgXq# “ g ¨ ιX# ¨ g´1

whose validity can be seen by expressing both sides of the equation in local coordinates.Thus

ιpAdgXq#pg ¨ αpXqq “ pg ¨ ιX#qαpXq

and we are done.

To define an equivariant cohomology on Ω8G pMq analogous to de Rham cohomology we wouldlike the product of two closed forms to be a closed form and we would like the differentialoperator to square to zero. This is the case (otherwise our choice of the equivariant differentialcomplex would not have been good).Lemma 3.5. For α, β P Ω8G pMq

1qdGpα^ βq “ dGα^ β ` p´1qdeg βα^ dGβ

2qd2G “ 0

The first equation says that dG is an anti-derivation.

Proof. 1) Follows as both d and ι are anti-derivations.

2) We will again solve this by direct computation. For α P Ω8G pMq we have

pd2GαqpXq “ pd` ιX#q

2pαpXqq “ pdιX# ` ιX#dqpαpXqq

14


since both d and ιX# squares to zero. By Cartan’s magic formula this last expression is equalto the Lie derivative along X#. So

pd2GαqpXq “ £X#pαpXqq. (27)

Now the equivariance of α comes into play. For any g P G we have

αpg ¨Xq “ g ¨ αpXq,

so specifically this is true for g “ exppsXq,

αpexppsXq ¨Xq “ exppsXq ¨ αpXq.

Taking the derivative at s “ 0 in this equation we obtain the corresponding infinitesimalactions of g. Following example (2.14) the right hand side of this equation is the Lie derivativeof αpXq along X#, which is what we have in (27) and want to show is zero. So if we can showthat the left hand side vanishes we are done. We have

d

dsαpexppsXq ¨Xq|s“0 “

d

dsαpAdexppsXqXq|s“0

and using the definition of the adjoint action

d

dsαpAdexppsXqXq|s“0 “

d

ds

ˆ

d

dtα`

exp psXq exp ptXq exp p´sXqq˘

|t“0

˙

|s“0

“d

ds

ˆ

d

dtα`

exp ptXq˘

|t“0

˙

|s“0 “ 0,

since the exponentials may be added to eliminate the s-dependency.

Now we have the basis for cohomology. Consider the subset of polynomial maps gÑ Ω˚pMqin Ω8G pMq. Clearly this set is closed under the wedge product so it forms a subalgebra. Wewill denote this set Ω˚GpMq (with a grading instead of the infinity superscript), whose elementscan be made very explicit using the standard identification

Ω˚GpMq “`

Crgs b Ω˚pMq˘G,

the set Crgs denoting the polynomial ring over g and the G-superscript denoting the subsetthat satisfies G-equivariance as maps g Ñ Ω˚pMq. We will call Ω˚GpMq the Cartan complexof equivariant cohomology. An element α P Ω˚GpMq may then be written

α “ φb ω

withαpXq “ φpXqω P Ω˚pMq,

and furthermoredGαpXq “ pd` ιX#qφpXqω “ φpXq

`

d` ιX#qω.

We see that we may then make the equivariant differential explicit. Let pXaqdimpGqa“1 be gener-

ators of the Lie algebra and pφaqdimpGqa“1 their dual elements in g˚,

φapXbq “ δab.

15


We have X “ caXa and likewise for the induced vector field X# “ caX

a#, so

dGαpXq “ φpXq`

d` caιXa#qω.

Noting φapXq “ ca we have

dGαpXq “ φpXq`

d` φapXqιXa#qω “

ˆ

p1b d` φa b ιXa#qφb ω

˙

pXq

so in factdG “ 1b d` φa b ιXa#

whose relaxed version isdG ” d` φaιXa .

This fits very well into the Cartan complex. We may consequently introduce a grading onΩ˚GpMq by

degGpαq “ 2 degP pφq ` degdRpωq,

i.e. we count twice the polynomial degree in Crgs plus the ordinary de Rham degree in Ω˚pMq.With this the equivariant differential becomes graded with degree 1, but only on the Cartancomplex. At last we have arrived at the following.Definition 3.6. The equivariant cohomology is

H˚GpMq “à

H iGpMq :“

à

kerpdG|ΩiGpMqqimpdG|Ωi´1

G pMqq.

Theorem 3.7 (Equivariant de Rham Theorem). If a compact Lie group acts on a compactmanifold, then the equivariant cohomology defined by the Cartan complex is isomorphic to thetopologically one defined in (24). [2]

Let us finally go into integration and see a few examples of the properties this cohomologyhas.Definition 3.8. Let α P Ω8G pMq and assume αpXq P Ω˚pMq is integrable over M . Then theintegration operator

ş

M : Ω8G pMq Ñ Cpgq is defined by

`

ż

Mα˘

pXq “

ż

M

`

αpXq˘

.

If αpXq does not contain a top form then this vanishes. For α “ φbω in the Cartan complexthis becomes

ż

Mα “ φ

ż

Mω P Crgs.

It will also be convenient to adopt the following notation. For an ordinary de Rham formα P Ω˚pMq we split it up into its different degrees by

α “ αr0s ` αr1s ` . . .` αrns (28)

so that for example αr0s is a function and αrns is a top-/volume form on M .

16

3 EQUIVARIANT COHOMOLOGY 3.2 Equivariant symplectic forms

Example 3.9. Ordinary de Rham theory has the typical property that the integral of aboundaryless domain descends to cohomology. This is due to Stokes’ theorem and is easilyproved:

ż

Mdω “

ż

BMω “ 0.

But the same is true for equivariant cohomology. The top form in an equivariantly exact formevaluated at any X, dGαpXq, must necessarily be of the form dαpXq as the interior productlowers the degree. Thus the same argument applies:

ż

MdGαpXqrns “

ż

MdαpXqrns “

ż

BMαpXqrns “ 0.

Example 3.10. Consider similarly the bottom form, i.e. the function part in Ω0pMq of anequivariantly exact form dGαpXq. This part must arise as the interior product of the 1-formin αpXq. Thus

dGαpXqr0s “ ιX#αpXqr1s.

At a fixed point of the action then, where the induced vector field is zero, this vanishes. Thusevaluating the bottom form at a fixed point is another map that descends to equivariantcohomology.Remark. In none of the examples above did we assume α was in the Cartan complex. Thusthe above results actually apply to the larger and coarser cohomology group of the wholeequivariant complex,

H8G pMq :“ kerpdG|Ω8G pMqqimpdG|Ω8G pMq

q.

We will later encounter forms that are not included in the Cartan complex, so this cohomologygroup will too be important.Example 3.11. Consider the integral of an equivariant form where we integrate both overthe manifold and the Lie algebra:

ż

gdX

ż

MαpXq.

Letting pXaqdimpGqa“1 again be generators of the Lie algebra we may write this as

ż

Rdca

ż

MαpcaX

aq.

3.2 Equivariant symplectic forms

Consider an equivariant differential form in the Cartan complex of degree 2, ω P Ω2GpMq.

Clearly this degree can only arise in two ways,

Ω2GpMq “ pC0rgs b Ω2pMqqG,‘pC1rgs b Ω0pMqqG

where the superscript in C˚rgs denotes the polynomial degree. Now, C0rgs ” C so the firstpart of the sum is just

pC0rgs b Ω2pMqqG ” pΩ2pMqqG,

so we may write an element of C0rgs b Ω2pMqqG as 1 b ω ” ω with ω a two-form invariantunder G,

g ¨ ω “ g´1˚ω “ ω.

17

3 EQUIVARIANT COHOMOLOGY 3.2 Equivariant symplectic forms

An element in the second part of the sum is φf P pC1rgs b Ω0pMqqG, with φ a polynom ofdegree one and f a function on M .

Let us see what conditions closedness poses on ω.

dGω “ 0 ðñ p1b d` φa b ιV aqp1b ω ` φfq “ 0

ðñ 1b dω ` φa b ιV aω ` φb df “ 0

ðñ 1b dω “ 0 and φa b ιV aω ` φb df “ 0

where in the last line we use that dω P Ω3pMq and ιV aω, df P Ω2pMq are of different degreeand must vanish independently. The first equality says that ω is closed in the de Rhamcohomology and the second equality is equivalent to, for all X P g

ιX#ω ` φpXqdf “ 0.

But this last expression may be written

dpφpXqfq “ ´ιX#ω

which is the criterion 1 in definition (2.20) of the moment map if we identify µ “ ´φf ,

µ : M Ñ g˚

µ : p PM Ñ ´φfppq P g˚.

But ´φf is also equivariant as in criterion 2 in the definition of the moment map and fur-thermore we have shown that ω is G-invariant and de Rham-closed. The only thing differentfrom the moment map definition of symplectic geometry is that we have said nothing aboutthe degeneracy of ω. We have shown the following.Theorem 3.12. For ω a symplectic form and µ its moment map,

ω :“ 1b ω ´ µ ” ω ´ µ

is a closed equivariant differential form.

Conversely, any ω P H2GpMq with ωr2s nondegenerate may be written as the difference of a

sympletic form and its moment map.

We will call the ω corresponding to a symplectic form ω the equivariant symplectic extension.This theorem gives us a mean of producing equivariant forms in the cohomology of a symplecticmanifold, as also

ω2, ω3, . . .

are equivariantly closed. Observe that unlike the case of the ordinary symplectic form ω theabove sequence does not terminate,

pω1, ω2, . . .q “ pω1, ω2, . . . , ωn, 0, 0, . . .q.

Consider also

eω :“ 1` ω `ω2

2` . . . (29)

18

4 EQUIVARIANT LOCALIZATION

which is a closed equivariant form as applying dG kills every term. Since µ commutes with ωwe have

eω “ eωe´µ “ e´µp1` ω `ω2

2` . . .`

ωn

n!q. (30)

Note in (29)-(30) that every finite partial sum is an element of the equivariant cohomologyH˚GpMq of the Cartan complex but the limital series is not, as the coefficient map on g is nota polynomial but a power series. However, eω is an element of the bigger cohomology groupH8G pMq from remark (3.1). This form is of great importance as it arises in the classical parti-tion function and is the focus of one of the big localization formulas in equivariant cohomology,the Duistermaat-Heckman formula. It will also appear later in our study of a supersymmetricquantum theory, in chapter 6. Thus considering the bigger cohomology group correspondingto the whole equivariant differential complex Ω8G pMq can be of importance.

We end this chapter with our returning example of a circle action on the sphere.Example 3.13. Let S1 act on S2. We know from example (2.21) that the symplectic extensionof the area form is

ω “ ω ´ µ “ sinφdφ^ dθ ´ pcosφ` λq,

and it is equivariantly closed. We can computeż

S2

ω “

ż

S2

ω “ 4π

as only the top form survives and thus this integral is constant as a function on g.

4 Equivariant Localization

There are a number of theorems of localization in equivariant cohomology, adjusted for differ-ent problems and assuming different properties of the Lie group action and the manifold. Wewill here prove one of the first and most fundamental localization theorems, catered for theaction of a circle group.Lemma 4.1. For the action of compact G on M , there is a Riemannian metric r on M thatis group invariant:

g ¨ r “ r.

Proof. The details are not important for us, but taking any metric r0 on M and averag-ing it over the group invariant Haar measure dg over G proves the lemma by construction.Explicitly

r “

ż

Gpg ¨ r0qdg

as can be done since G is compact.Theorem 4.2 (Berline, Vergne 1982; Atiyah, Bott 1984). Assume the following: M iscompact, orientable, boundaryless and of even dimension n; G is compact, connected and1-dimensional; the fixed points of the action are isolated; α is a closed equivariant differentialform. Then

ż

Mα “ p´2πqn2

ÿ

k

αr0spykq1

a

det BiV j |yk

,

where V is the vector field induced by a nonzero V P g and the sum is over the fixed points yk.

19


Remark. By examples (3.9)-(3.10) both the left- and right side of the above equation descendto cohomology (assuming the theorem to be true, any of them would have had to imply theother) so the theorem too descends to equivariant cohomology.

Proof. We divide the proof into several steps.Claim (Localization principle). Assume β satisfies d2

Gβ “ 0. Then the expression

Zpsq :“

ż

Mαe´sdGβ

is independent of s.

Since M is boundaryless the integral of an exact form is zero - see example (3.9). Thus if wecan show αp1´ e´sdGβq is an exact form we are done as this will imply

ż

Mα “

ż

Mαe´sdGβ,

and the right hand side is independent of s. But

αp1´ e´sdGβq “ αÿ

i“1

p´sdGβqiqi!

“ αdGÿ

i“0

´sβp´sdGβqipi` 1q! “ dG

ˆ

αÿ

i“0

´sβp´sdGβqipi` 1q!

˙

using that dGβ is a closed form by assumption. 4

Observe that the integral we want to calculate is Zp0q for the function Z defined above, butwe may choose any β and any s that we deem Zpsq to be calculable for, as Z should renderthe same value independently. So let us abuse this.Claim (Choice of β). Take V as in the theorem and r as in the preceding lemma (4.1). Wemay then let

β :“ rpV, ¨q “ rµνVµdxν .

We need to show d2Gβ “ 0. We saw in the previous chapter that the equivariant operator

squares to the Lie derivative. Thus the condition d2Gβ “ 0 may be stated as: for any vector

field W induced by the Lie algebra we have

£Wβ “ 0.

But the Lie group is one dimensional so any two elements in the Lie algebra must differ byonly a constant scalar and the same applies to the corresponding vector fields. Thus the abovestatement is equivalent to £Wβ “ 0 for some nonzero W P g, so specifically for W “ V . Nowr is G-invariant so by corollary (2.16.1) it is infinitesimally invariant,

£V r “ 0.

But the Lie derivative of V µ along V is also zero so we must have

£V β “ 0

(we could also have expressed the above locally in coordinates to see LV β cancels). 4

20


Claim (Localization at stable points). With this choice of β, the integral

Zpsq “

ż

Mαe´sdGβ

only depends on finitely many, arbitrarily small neighborhoods of the stable points of theaction.

Our choice of β gives

dGβ “ pd` ιV qrµνVµdxν “ rµνV

µV ν ` BγprµνVµqdxγdxν ,

soZpsq “

ż

Mαe´sprµνV

µV ν`BγprµνV µqdxγdxνq.

But the first termrµνV

µV ν P Ω0pMq

is just a function which commutes with forms of higher degree so

Zpsq “

ż

Mαe´srµνV

µV νe´sBγprµνVµqdxγdxν .

Rewriting this integral using proposition (2.24) as an integral over the odd tangent bundle,we obtain

Zpsq “

ż

ΠTMdnxdnηαpx, ηqe´srµνV

µV νe´sBγprµνVµqηγην .

Since r is a Riemannian metric it is positively definite and

rµνVµV ν ě 0

with equality iff V “ 0, i.e. where the action is infinitesimally invariant. Since the Lie groupis assumed to be compact and connected though, infinitesimal invariance is the same as groupinvariance by corollary (2.16.1). Therefore, in the limit sÑ8 we have

e´srµνVµV ν Ñ 0

except at the stable points of the action. But

αpxq “ÿ

fi1,...,ikpxqdxi1 . . . dxik .

is bounded (it is a form on a compact manifold), so at any point not stable

αpx, ηq “ÿ

fi1,...,ikpxqηi1 . . . ηik.

gets killed by the exponential function. Also the other factor

e´sBγprµνVµqηγην

is just a polynomial in the η which cannot compete with the exponential factor. Hence in thelimit sÑ8 the whole integrand goes to zero except at the stable points of the action, whereV “ 0.

We conclude that only neighborhoods arbitrarily small around the stable points will contributeto the integral. By assumption these are isolated and since M is compact they are finitelymany. 4

21


Claim (Contribution of a single point). Say that a stable point of the action is y. Thecontribution to the integral of this point is

p´2πqn2αpy, 0qq1

a

det BiV j.

Let us denote the finite number of stable points yk. Then by the above claim we may findneighborhoods Myk of these points that are disjoint and the restriction of the integrand tothese domains satisfies

Zpsq “

ż

ΠTMdnxdnηαpx, ηqe´srµνV

µV νe´sBγprµνVµqηγην

“ÿ

k

ż

ΠTMyk

dnxdnηαpx, ηqe´srµνVµV νe´sBγprµνV

µqηγην .

Let us consider one of these integrals over an arbitrarily small My corresponding to the stablepoint y. Expanding rµνV µV ν at y “ py1, . . . , ynq and using V pyq “ 0 we have

rµνVµV ν |x “ rµνV

µV ν |y ` BirµνVµV ν |ypx

i ´ yiq

` BijrµνVµV ν |ypx

i ´ yiqpxj ´ yjq Òpx3q

“ rµνBiVµBjV

ν |ypxi ´ yiqpxj ´ yjq Òppx´ yq3q,

as only terms containing differentiations of both the V -components do not vanish at y.

Let us now employ the change of coordinates

xi Ñ xi,

ηi Ñ ηi,

where

xi “ yi `xi?s

ηi “ηi?s.

This gives

rµνVµV ν |

py` x?sq“ rµνBiV

µBjVν |y

xixj

sÒp

x3

?s3 q

ande´srµνV

µV ν |x “ e´rµνBiV

µBjVν |yxixjÒp

x3?sq

and in the limit sÑ8 we have

limsÑ8

e´srµνVµV ν |x “ e´rµνBiV

µBjVν |yxixj

The other exponential factor in the new coordinates becomes

´sBγprµνVµq|xη

γην “ ´BγprµνVµq|py` x?

sqηγ ην

22


and in the limit sÑ8 we have

´BγprµνVµq|py` x?

sqηγ ην Ñ ´BγprµνV

µq|yηγ ην

independent of the x- and x-coordinates. Also, V µ|y “ 0 so

´BγprµνVµq|y “ ´rµνBγV

µ|y.

The last factor in the integrand αpx, ηq has

αpx, ηq “ αpy `x?s,η?sq

which in the limit sÑ8 becomes

αpx, ηq Ñ αpy, 0q.

Using the formulas in equation (20) and proposition (18) for the changes in measure in thetransformations we have

dnx “ p?sqndnx

anddnη “ 1p

?sqndnη,

so these effects of the coordinate changes cancel. Finally, the integrand inż

ΠTMy


µqηγην

is bounded and we may use the dominated convergence theorem and all our assertions on thelimits of the integrand as sÑ8 to conclude

limsÑ8

ż

ΠTMy


µqηγην

“

ż

ΠTMy

limsÑ8


µqηγην

“

ż

ΠTMy

dnxdnηαpy, 0qe´rµνBiVµBjV

ν |yxixje´rµνBγVµ|y ηγ ην .

Now this integral may be separated into two Gaussian integrals over My and the Grassmannalgebra Λ respectively as

αpy, 0q

ż

My

dnxe´rµνBiVµBjV

ν |yxixjż

Λdnηe´rµνBγV

µ|y ηγ ην .

These we can evaluate. Denoting the coefficient matrix for the quadratic form in the xi

Hij :“ rµνBiVµBjV

ν |y

and the suiting coefficient matrix for the quadratic form in the ηi

Kij :“ 2rµjBiVµ|y

23


we obtain, using the Gaussian integration formulas in equation (21) and proposition (19),ż

My

dnxe´rµνBiVµBjV

ν |yxixj “

ż

My

dnxe´Hij xixj “ πn2

1a

detHij

andż

Λdnηe´rµνBγV

µ|y ηγ ην “

ż

Λdnηe´

12Kij ηiηj “ p´1qn2

a

detKij ,

so putting these factors together the integral becomes

αpy, 0qp´πqn2a

detKija

detHij.

The last detail in the claim comes from noting that

Hij “1

2p2rµνBiV

µqBjVν |y “

1

2KiνBjV

ν |y

is the product of two matrices so

detHij “1

2ndetKij det BiV

j |y.

This gives the integral contribution

p´2πqn2αpy, 0q1

a

det BiV j |y.

4

Discarding of the Grassmann variables as in αpy, 0q just leaves the function part of α, soαpy, 0q “ αr0spyq. The complete proof now follows from the last claim by summing over allthe stable points of the action,

ż

Mα “ p´2πqn2

ÿ

k

αr0spykq1

a

det BiV j |yk

.

Remark. An integralş

M α is completely independent of anything but the topform in α - therest vanishes upon taking the integral. But for α to be closed equivariantly, all its componentforms of lower degree come into play, related to one another by the condition pd` ιV qα “ 0.In the end, the BVAB localization formula tells us that we can read the integral value of aclosed form α from only its zero-form part, the function, at a certain set of points. Amazing!

Example 4.3. We may return to study our old example of the circle action on S2. Saywe wish to integrate the standard area form ω “ sinφdφ ^ dθ (which we have done a fewtimes already - it is 4π) using the theorem just proved. From chapter 3.2 we know that theequivariant extension ω is a closed equivariant form, and since the top form is the same in ωand ω they have the same value when integrated over M ,

ż

S2

ω “

ż

S2

ω,

24

5 EQUIVARIANT COHOMOLOGY IN QUANTUM MECHANICS

independently of where in g we evaluate it. But this latter integral fulfils all requirements inthe BVAB theorem just proved. Thus its integral should localize onto the two fixed pointsof the rotational circle action, which by inspection we see is the south- and north pole. Thefunction part of ω is ´µ “ ´z where we set the constant parameter to zero, and passing toeuclidean coordinates which include the two poles we obtain

ż

S2

ω “ p´2πq22ÿ

np,sp

´z1

a

det BipxBy ´ yBxq“ ´2πp´1´ 1q “ 4π.

5 Equivariant Cohomology in Quantum Mechanics

Equivariant cohomology is interesting in its own right, as the previous chapters have shown,but let us see how it may be relevant to quantum mechanics and even further to supersym-metric quantum mechanics. We will give two motivational ideas correspondingly, and in thefinal chapter we combine these to motivate a known result about a specific theory, namelythat the partition function localizes on specific path configurations. To begin with we aregoing to need the Feynman formalism of quantum mechanics, that of path integrals.

5.1 Path integral formalism

Let us for clarity assume we have only one particle that is constrained to move within onedegree of freedom. The Feynman formalism then states that given our system in a specificconfiguration at an initial time |qi, tiy the probability that the system will be in anotherconfiguration at a time later |qf , tf y is obtained by summing the probability of every possiblepath the system can take between these points. The probability of a single path is in turndetermined by integrating the exponential of the action, where the action is the classicalone

S “

ż

Ldt.

Summing over every conceivable path results in an infinite dimensional integral - not welldefined by any means - but which may still be calculable to give correct predictions. Weshowed in chapter 2.2 that a mechanical system may be identified with a symplectic manifoldpM,ωq and that its dynamics follow

9f “ i~rf , Hsω

for the Hamiltonian H ” H and an observable f ” f . It is shown in for example [4] that thepath integral described above from |qi, tiy to |qf , tf y is

xqf , tf |qi, tiy “

ż

rdq dps exppi

~Srq, psqδpqptiq ´ qiqδpqptf q ´ qf q

where the Feynman measure rdq dps denotes summing over all paths and is defined by aninfinite time slicing

rdq dps ” ΠtPr0,T sdpptq

2π~dqptq.

25

5 EQUIVARIANT COHOMOLOGY IN QUANTUM MECHANICS 5.2 Loop space

The action is

S “

ż tf

ti

9qptqpptq ´Hpq, pqdt.

The last factor δpqiptiq ´ qiiqδpqiptf q ´ qif q ensures we have the correct start- and end points.

It is conventional to set ~ “ 1 since it is irrelevant to the analysis and we will also assumeti “ 0, tf “ T . What one usually wants to calculate in quantum mechanics is the partitionfunction, which expressed using the Feynman path integral formalism becomes

ZpT q “

ż

rdq dps exppiSrq, psqδpqptiq ´ qptf qq,

i.e. the integral over all closed paths of period T , taking all starting points into account [4].Finally generalizing this to 2n coordinates is straight forward. Recalling the Liouville volumeform (12) we have in arbitrary coordinates

dq1 . . . dqndp1 . . . dpn “1

n!ωn “

b

detωijpxqdx1 . . . dx2n

so over a curve of period T we may write the general partition function as

ZpT q “

ż

rd2nxsΠtPr0,T s

b

detωijpxptqq eiSrxsΠiδ

`

xiptiq ´ xiptf q

˘

.

5.2 Loop space

For M a manifold, we may introduce the space of loops, i.e. closed curves, on M of a specificperiod T . We denote this space LM , so that an element x P LM is a curve on M withxp0q “ xpT q. Loop space is an infinite dimensional manifold, actually an infinite dimensionalfiber bundle overM with fiber consisting of all loops starting at a point [4]. With this we maywrite the partition function as an integral over loop space of phase space M :

ZpT q “

ż

LMrd2nxsΠtPr0,T s

b

detωijpxptqqeiSrxs. (31)

The connection of equivariant localization to quantum mechanics may now be made appar-ent. Loop space naturally holds a Lie group action by identifying loops with a circle group.Explicitly, if we rescale time so that T “ 2π, we may let τ P S1 act on a coordinate xaptqby

τ ¨ xaptq “ xapt` τq,

and we can define this action diagonally on as many of the coordinates as pleases. This allowsfor an equivariant cohomology on loop space, and one can hope to be able to calculate thepartition function in (31) with the help of a localization theorem as the one proved previously,in chapter 4.

So loop space will be important to us; let us list some of its properties. A differential k-formis

α “1

k!

ż T

0dt αi1...ikpxptqqdx

i1ptq . . . dxikptq,

26

5 EQUIVARIANT COHOMOLOGY IN QUANTUM MECHANICS 5.2 Loop space

and the exterior derivative dL and the contraction operator ι are respectively

dL “

ż T

0dt dxaptq

δ

δxaptq,

ιV “

ż T

0dt V arx; tsptq

δ

δxaptq.

So a symplectic form on loop space is a 2-form

Ω “1

2

ż T

0dt Ωµνpxptqqdx

µptqdxνptq

which is closed:dLΩ “ 0,

and its moment map with respect to some Lie group induced vector field V a is defined by

dLµVL “ ιV Ω,

where as a 0-form

µVL “

ż T

0dtµV rx; ts.

If pM,ωq is symplectic this canonically induces a symplectic form Ω on loop space by

Ωµν “ ωµν .

One can then see that µ is a moment map for pM,ωq if and only if

µL :“

ż T

0dtµ (32)

is a moment map for pLM,Ωq.

A final observation is the following. The factora

detωijpxq is what results from a Grassmannintegration of the Gaussian exp

`

12η

iωijpxqηj˘

as in proposition (2.23). Therefore

ΠtPr0,T s

b

detωijpxptqq “ ΠtPr0,T s

ż

Λexp

`1

2ηiptqωijpxptqqη

jptq˘

“

ż

Λexp

`

ż T

0dt

1

2ηiptqωijpxptqqη

jptq˘

.

The canonically induced symplectic form Ω with the identification dxi “ ηi has

Ωrx, η; ts “1

2

ż T

0dt ωµνpxptqqη

µptqηνptq

and thus, again using proposition (2.24) to rewrite the partition function as an integral overthe odd tangent bundle of LM ,

Z “

ż

LMrd2nxsΠtPr0,T s

a

detpωpxptqqqeiSrxs “

ż

ΠLMrd2nxsrd2nηseipSrxs`Ωrx,ηsq. (33)

The odd tangent bundle ΠLM is called super loop space and the exponential

A “ S ` Ω,

occurring in the above partition function is called the augmented action on ΠLM .

27

5 EQUIVARIANT COHOMOLOGY IN QUANTUM MECHANICS 5.3 Supersymmetry

5.3 Supersymmetry

Supersymmetry is for our purpose a transformation that for a given system relates bosonic,commutating fields xa with fermionic, anticommutating fields ψa, where the latter are rep-resented by Grassmannian variables. In this paper, to consider systems where equivariantlozalization may be applicable, we will study a system with the following assumed supersym-metric transformations: one fermionic δf - and one bosonic δb transformation acting on ourfields as well as a series of auxiliary bosonic fields φa by

δfxa “ ψa (34)

δfψa “ δbx

a (35)δfφ

a “ 0. (36)

We will call a system of such supersymmetric transformations a BRST symmetry. To seethe connection to equivariant cohomology, consider the equivariant operator dG “ d ` ιV oncoordinates xa:

dGxa “ dxa

dGdxa “ £V x

a “ V a

dGVaTa “ 0.

Recalling the identification dxa ” ψa which we have already used a number of times, we seethat the fermionic transformation in a BRST-theory is truly similar to an equivariant differ-ential. Pursuing this further, the auxiliary fields φa get identified with Lie algebra elementsand the bosonic supersymmetry δb gets identified with the Lie derivative along V . We maythen see the underlying Lie group action as defined by its induced vector fields via the Liederivative in either of

£V xa “ δ2

fxa “ δbx

a.

If the Lie group is compact and connected (e.g. a circle action) then this is well defined, asfollows from the surjectivity of the exponential map.

The partition function for a system described above is of the form

Z “

ż

rdxasrdψasrdφaseiSrxa,ψa,φas

with the integration over super loop space. We will in the next chapter combine the two moti-vating arguments given in this chapter to calculate such a partition function using equivariantlocalization.

28

6 EQUIVARIANT LOCALIZATION IN SUPERSYMMETRIC QUANTUM MECHANICS

6 Equivariant Localization in Supersymmetric Quantum Me-chanics

It is shown in [7] that for a system with one bosonic field x and two fermionic fields ψ, ψ theaction and the partition function are respectively

A “

ż

dt

ˆ

1

29x2 ´

1

2pW 1pXqq2 ` ψriBt ´W

2pxqsψ

˙

Z “

ż

rdxsrdψsrdψseiArx,ψ,ψs,

and furthermore the action is invariant under the SUSY transformations

δx “ iεψ ` iεψ

δψ “ ´εp 9x´ iW 1pxqq

δψ “ ´εp 9x` iW 1pxqq.

Starting from this, we are interested in the specific case of BRST symmetry as in (34)-(36),suggesting ε “ 0, ε “ ´i:

δx “ ψ

δψ “ 0

δψ “ i 9x´W 1pxq.

The above does not completely resemble a theory with BRST symmetry as a commutatingfield with differential ψ is lacking. We will therefore add such a field π without changing theaction. The system is then described by

Z “

ż

rdxsrdπsrdψsrdψseiArx,ψ,ψs (37)

A “

ż

dt

ˆ

1

29x2 ´

1

2pW 1pXqq2 ` ψriBt ´W

2pxqsψ

˙

,

but letting

S “

ż

dt

ˆ

1

29x2 ´

1

2pW 1pXqq2

˙

andΩ “

ż

dtdt1dπptqriBt ´W2pxqsδpt´ t1qdxpt1q,

we see that the partition function in (37) may actually be interpreted as a partition functionon phase spaceM “ px, πq written as an integral over super loop space with augmented actionA. This will be key later in applying equivariant localization. Note also that Ω is symplecticon LM as it is obviously closed being a top form.

29


We now add an auxiliary field D which we want to identify with the Lie algebra, i.e. we wantthe Lie derivative to give us D as in (35) and hence we consider

δf “ δ (38)δfx “ ψ (39)δfψ “ 0 (40)δfπ “ ψ (41)δf ψ “ D. (42)

We can add such a field D if it can be integrated out of the action to return i 9x´W 1pxq. Thisleads us to consider the action

S1 “

ż

dt`

pí 9x`W 1pxqqD `1

2D2

˘

with thenż

rdDsS1 “

ż

dt`

pí 9x`W 1pxqqpi 9x´W 1pxqq `1

2pi 9x´W 1pxqq2

˘

“

ż

dt`1

29x2 ´

1

2pW 1pxqq2 ´ i 9xW 1pxq

˘

which is just what we got in our action S except for the last term, but

i 9xW 1pxq “d

dtiW pxq

is an exact derivative, and since we are only considering closed paths the surface terms fromintegrating this cancel. Therefore we may indeed insert the auxiliary field D and consider theaction S Ñ S1 to obtain

Z “

ż

rdDsrdxsrdπsrdψsrdψseiArD,x,ψ,ψs, (43)

A “

ż

dt

ˆ

pí 9x`W 1pxqqD `1

2D2 ` ψriBt ´W

2pxqsψ

˙

.

We will now show that the supersymmetry transformations in (38)-(42) do in fact define acircle action on LM as described in chapter (5.2). The group action should induce

£V pDqx “ 0

£V pDqπ “ D

where D is Lie algebra valued so the induced vector field by S1 is

V “ DB

Bπ.

The action is then, for τ P S1,

τ ¨ pxptq, πptqq “ pxptq, πpt` τqq.

30


Now that we have established the role of δf as an equivariant differential operator, let us seehow equivariant localization can help calculate the partition function (43). It will turn outthat A is the equivariant symplectic extension of

Ω “

ż

dtdt1dπptqriBt ´W2pxqsδpt´ t1qdxpt1q

on LM evaluated at D in the Lie algebra. The moment map µL satisfies

dLµπL “ ιBπΩ

or equivalently on phase space M , as noted in (32),

dµπ “ ιBπ`

dπriBt ´W2pxqsdx

˘

so

Bµπ

Bx“ iBt ´W

2pxq

Bµπ

Bπ“ 0

and we haveµπ “ i 9x´W 1pxq

ignoring the constant of integration. So the equivariant symplectic extension is

Ω “ Ω´ µL “

ż

dt

ˆ

´ i 9x`W 1pxq ` dπptqriBt ´W2pxqsdxptq

˙

but evaluating this at D gives, as in example (3.11),

ΩpDBπq “ Ω´DµπL “

ż

dt

ˆ

p´i 9x`W 1pxqqD ` dπptqriBt ´W2pxqsdxptq

˙

.

Now comparing this with A, we see that apart from the term 12D

2 this is exactly the bosonicversion of the integrand A over LM , again using proposition (2.24). Thus, rewriting usingthis proposition, we have

Z “

ż

rdDsrdxsrdπsrdψsrdψseiArD,x,ψ,ψs

“

ż

rdDs

ż

LMexpri

`

Ω´DµπL `

ż

dtp1

2D2q

˘

s

“

ż

rdDs

ż

dt exppi1

2D2q

ˆż

LMeiΩ

˙

pDBπq

where in the last line we divide the integral into one outer integral over the Lie algebra and oneinner integral of Ω over LM , as we did in example (3.11). Finally the equivariant symplecticextension Ω is closed over LM , so we see that the partition function beautifully has windup as the integral of an equivariantly closed form, just as we studied in chapter 4 only nowinfinite dimensional. We could hope to calculate this using some generalized version of theBVAB formula. Blindly disregarding the small detail of infinite dimensionality for a minute

31

7 CONCLUSION

and using the BVAB formula as proved in this paper, this integral should localize on the fixedpoints of the action, where D “ 0. Recalling D “ i 9x´W 1pxq we then obtain that the partitionfunction localizes on those paths satisfying

i 9x´W 1pxq “ 0.

We may strengthen this statement. Since 9x and W 1pxq are both real quantities they must infact both vanish at the localized paths,

9x “ 0

W 1pxq “ 0

so the localization is by the first equation onto the constant paths over phase space, whichis just phase space! And the second equation further reduces the localization to those pointswith W 1pxq=0. This result is very strong. We have reduced an infinite dimensional integralto a sum over those points in finite dimensional phase space that are critical points of thepotential - a set that is typically finite!

This result copies that in [5], where a different approach is used to obtain the same localizationresult. Thus applying localization to this system does in fact yield the correct result.

7 Conclusion

The Cartan model makes equivariant cohomology, that intuitively was to be the de Rhamcohomology modulo the group action, very explicit. Although this was the cohomology we setout to define we saw that there are cases when a bigger cohomology group, not restricted topolynomial maps on the Lie algebra as coefficients, should be considered. The properties weshow in this paper descend to both cohomology groups. Furthermore, we see that symplecticgeometry - whose role was completely necessary in creating the geometrical framework fordynamics in which cohomology could be applied - has a natural and useful role in equivariantcohomology via the moment map. At the end of our study of equivariant cohomology weprove the well known BVAB localization theorem.

Regarding the application to quantum mechanics we see that loop space readily permits acircle group action and that the specific case of BRST supersymmetry when interpreted asan equivariant differential operator can define a Lie group action. These two argumentsenable equivariant cohomology in quantum mechanics. Lastly, taking a standard model ofsupersymmetry and applying the above ideas we rewrite the partition function as an integralof equivariant cohomology which, using the BVAB theorem, then localizes onto a small setof path configurations. This is a rederivation of a known result that shows how equivariantlocalization with success may be used in quantum mechanics.

32

References

[1] L.Akant hep-th/0505242

[2] V. Guillemin, S .Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag, 1999.

[3] M. Nakahara Geometry, Topology and Physics, MPG books Ltd, Cornwall, 2nd edition,2003

[4] R. J. Szabo hep-th/9608068

[5] D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Physics reports, vol 209, 1991

[6] R. Berndt An Introduction to Symplectic Geometry, AMS, 2001

[7] F. Cooper, A. Khare, U. Sukhatme hep-th/9405029,

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