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TheM athai-Quillen Form alism and

TopologicalField Theory�

M atthiasBlauy

NIKHEF-H

P.O.Box 41882,1009 DB Am sterdam

The Netherlands

M arch 9,1992

A bstract

These lecture notes give an introductory account ofan approach to

cohom ological�eld theory due to Atiyah and Je�rey which is based on

the construction ofG aussian shaped Thom form sby M athaiand Q uillen.

Topics covered are: an explanation ofthe M athai-Q uillen form alism for

�nitedim ensionalvectorbundles;thede�nition ofregularized Eulernum -

bers of in�nite dim ensionalvector bundles; interpretation of supersym -

m etricquantum m echanicsastheregularized Eulernum berofloop space;

theAtiyah-Je�rey interpretation ofDonaldson theory;theconstruction of

topologicalgauge theories from in�nite dim ensionalvector bundles over

spacesofconnections.

NIKHEF-H/92-07

�Notes oflecturesgiven atthe K arpacz W interSchoolon ‘In�nite Dim ensionalG eom etry

in Physics’(17 -27 February 1992).ye-m ail:t75@ nikhefh.nikhef.nl,22747::t75

1

C ontents

1 Introduction 2

2 T he M athai-Q uillen Form alism 5

2.1 TheEulernum berofa �nitedim ensionalvectorbundle :::::: 5

2.2 TheThom classand theM athai-Quillen form ::::::::::: 8

2.3 TheM athai-Quillenform alism forin�nitedim ensionalvectorbundles 12

3 T he EulerN um berofLoop Space and Supersym m etric Q uantum

M echanics 15

3.1 Loop spacegeom etry ::::::::::::::::::::::::: 15

3.2 Supersym m etric quantum m echanics :::::::::::::::: 16

3.3 TheM athai-Quillen form from supersym m etric quantum m echanics 21

4 T he Euler N um ber ofVector B undles over A =G and Topological

G auge T heory 21

4.1 Geom etry ofgaugetheories ::::::::::::::::::::: 22

4.2 TheAtiyah-Je�rey Interpretation ofDonaldson theory :::::: 24

4.3 Flatconnectionsin two and threedim ensions ::::::::::: 28

R eferences 32

1 Introduction

Topological�eld theory has been a lively area for research ever since the ap-

pearance ofthe sem inalwork by W itten [1,2,3]a few years ago. Activity in

the�eld increased when theobservation wasm ade[4,5]thattopologicalgravity

in two dim ensionsisclosely related to two-dim ensionalquantum gravity and its

description in term sofrandom m atrix m odels.Severalreviewsofthesubjectare

now available1.

Iwilltry to com plem entthese existing reviewsby focussing on an approach

to topological�eld theory based on theconstruction by M athaiand Quillen [10]

ofGaussian shaped Thom form sfor�nitedim ensionalvectorbundles.Thisvery

elegant approach is due to Atiyah and Je�rey [11]who realized that topolog-

ical�eld theory could be regarded as an in�nite dim ensionalgeneralization of

thisconstruction. There are severaladvantagesofadopting thispointofview.

Firstofall,itprovidesan a prioriexplanation ofthefactthat�nitedim ensional

topologicalinvariants can be represented by functionalintegrals,the hallm ark

oftopological�eld theory. M oreover,ithas the charm ing property ofgiving a

1See[6,7,8]foran accountoftherelation am ong topologicalgravity,m atrix m odels,inter-

section theoryon m odulispace,and integrablem odels,and [9]forageneralreview oftopological

�eld theory.

2

uni�ed description ofallkindsof(cohom ological)topological�eld theoriesand

supersym m etric quantum m echanics. Thishasthe added bonusofm aking this

approachquiteelem entaryasitallowsonetodevelop them ainideasinaquantum

m echanicalsetting and to then transfer them alm ost verbatim to �eld theory.

Lastly,it also provides som e insight into the m echanism ofthe localization of

path integralsin supersym m etricquantum m echanicsand topological�eld theo-

ry.

To those already fam iliarwith the subject,these lectureswillhopefully pro-

vide a new and perhapsenlightning perspective on topological�eld theory. At

thesam etim ethey should,ideally,constitutean elem entary introduction to the

subjectrequiring no priorknowledgeofthe�eld and littlem orethan som ebasic

di�erentialgeom etry and theability to perform Gaussian integrals.

The recurrent them e in these notes willbe the Euler num ber ofa vector

bundle. In order to understand the basic idea ofthe Atiyah-Je�rey approach,

letusthereforerecallthatclassically thereexisttwo quitedi�erentprescriptions

for calculating the Euler num ber �(X ) � �(TX ) of(the tangent bundle of)a

m anifold X . The �rst is topologicalin nature and instructs one to choose a

vector�eld V on X with isolated zerosand to countthesezeroswith signs(this

isthe Hopftheorem ). The second isdi�erentialgeom etric and represents�(X )

astheintegraloverX ofa density (top form )er constructed from thecurvature

ofsom e connection r on X (the Gauss-Bonnet theorem ). Likewise,the Euler

num ber�(E )ofsom eothervectorbundleE overX can bedeterm ined in term s

ofeithera section s ofE ora connection r on E .

A m ore generalform ula,obtained by M athaiand Quillen [10],interpolates

between thesetwo classicalprescriptions.Itrelieson theconstruction ofa form

es;r (E )which dependson both a section s and a connection r . Thisform has

theproperty that

�(E )=

Z

X

es;r (E )

foralls and r . M oreover,thisequation reducesto the HopforGauss-Bonnet

theorem forappropriate choice ofs (forisolated zerosto the form erand to the

latterfors= 0).

W hatAtiyah and Je�rey [11]pointed outwasthat,although er andR

X er do

notm ake sense forin�nite dim ensionalE and X ,the M athai-Quillen form es;r

can be used to form ally de�ne regularized Euler num bers�s(E )ofsuch bundles

by

�s(E ):=

Z

X

es;r (E )

forcertain choicesofs.Although notindependentofs,thesenum bers�s(E )are

naturally associated with E fornaturalchoicesofsand arethereforelikely to be

oftopologicalinterest.

Itispreciselysuch arepresentation oftopologicalinvariants(inanon-technical

3

sense2)by functionalintegralswhich isthecharacteristicproperty oftopological

�eld theories,and which could also be taken astheirde�nition. Thissuggests,

thatcertain topological�eld theoriescan beinterpreted orobtained in thisway.

It willbe the m ain aim ofthese notes to explain that this is indeed the case

for the cohom ologicaltheories (i.e.not Chern-Sim ons theory and its siblings).

The m odelswe considerexplicitly are,in addition to supersym m etric quantum

m echanics,Donaldson theory[1]and varioustheoriesof atconnectionsdiscussed

e.g.in [13,14,15]and [16,17,18].Thisfram ework is,however,broad enough to

includetopologicalsigm a m odels,twisted m inim alm odels,and theircoupling to

topologicalgravity aswell(see[19,20]).

Thefollowingnotesconsistofthreesections,dealingwith theM athai-Quillen

form alism ,supersym m etric quantum m echanics, and topologicalgauge theory

respectively.Each section beginswith abriefreview oftherequired m athem atical

background.Thussection 2.1 recallstheclassicalexpressionsfortheEulerclass

and Eulernum berofa�nitedim ensionalvectorbundle.Forourpresentpurposes

the Euler num ber ofa vector bundle is best understood in term s ofits Thom

class and section 2.2 exlains this concept. It also contains the construction of

theGaussian shaped Thom form ofM athaiand Quillen and itsdescendantses;r .

Section 2.3dealswith theapplication oftheM athai-Quillen form alism toin�nite

dim ensionalvector bundles and their regularized Euler num ber and introduces

theexam plesto bediscussed in m oredetailin thesubsequentsections.

Section 3.1 contains the bare essentials ofthe geom etry ofthe loop space

LM ofa m anifold M necessary to apply the M athai-Quillen form alism to its

tangent bundle. Section 3.2 exlains how supersym m etric quantum m echanics

can be interpreted as de�ning or arising as a path integralrepresentation of

theregularized Eulernum berofLM .Som erelated resultslikethepath integral

proofsoftheGauss-Bonnetand Poincar�e-Hopftheorem sarereviewed in thelight

ofthisderivation.In section 3.3 itisshown thatthe�nite dim ensionalM athai-

Quillen form can,in turn,bederived from supersym m etric quantum m echanics.

Section 4.1dealswith thegeom etryofgaugetheories.W ederivean expression

forthecurvatureform oftheprincipal�bration A ! A =G and givea form ulafor

theRiem ann curvaturetensorofm odulisubspacesM � A =G.W ealsointroduce

thosein�nitedim ensionalbundleswhich willenterintothesubsequentdiscussion

oftopologicalgaugetheory.In section 4.2 itisshown thatthepartition function

ofDonaldson theory can be interpreted as the regularized Euler num ber ofa

bundle ofself-dualtwo-form s over A =G. It also contains a briefdiscussion of

2W hat is m eant by ‘topological’in this context is the invariance ofnum bers like �s(E )

underdeform ationsofcertain ofthe data entering into itscalculation. Itisin thissense that

theDonaldson invariantsoffour-m anifolds[12],which ariseascorrelation functionsofthe�eld

theory considered in [1],aretopologicalasthey areindependentofthem etricwhich entersinto

the de�nition ofthe instanton m odulispace. They are,however,nottopologicalinvariantsin

them athem aticalsenseasthey havetherem arkableproperty ofdepending on thedi�erentiable

structureofthe four-m anifold.

4

som e propertiesoftopological�eld theoriesin general,aswellassom e rem arks

on the interpretation ofobservables in the present setting. Topologicalgauge

theoriesof atconnectionsin twoand threedim ensionsarethesubjectofsection

4.3.In particular,in 3d wesketch theconstruction ofa topologicalgaugetheory

representing theEulercharacteristicofthem odulispaceof atconnections;once

directly from thetangentbundleofA =G and oncefrom supersym m etricquantum

m echanicson A =G.W ealso constructa two-dim ensionalanalogueofDonaldson

theory representing intersection theory on m odulispacesof atconnections.

Thebasicreferencesforsection 2.1 and 2.2 areBottand Tu [21]and M athai

and Quillen [10].Forsection 2.3 see[11]and [17].Them ain resultofsection 4.2

isdueto Atiyah and Je�rey [11],and a detailed discussion ofDonaldson theory

[1,12]can be found in [9,pp.198-247]. Sections3.2,3.3 and 4.3 are based on

jointworkwith GeorgeThom pson [16,17,18].Furtherreferencescan befound in

the textand furtherinform ation on topological�eld theory in the cited reviews

and thelecturesofDanny Birm ingham [22]atthisSchool.

2 T he M athai-Q uillen Form alism

In section 2.1 we wilrecallsom e wellknown factsand theorem sconcerning the

Eulerclassand the Eulernum berofa �nite dim ensionalvectorbundle E . For

our present purposes the Euler class is m ost pro�tably understood in term s of

theThom classofE and wewilladoptthispointofview in section 2.2.Therewe

also introduce and discuss atsom e length the M athai-Quillen form alism which

provides,am ongotherthings,aconcretedi�erentialform realization oftheThom

class. In section 2.3 we explain how the M athai-Quillen form alism can be used

to de�necertain regularized Eulernum bersofE when E isin�nitedim ensional.

W ewillalso introducetheexam ples(supersym m etricquantum m echanics,topo-

logicalgaugetheory)which willthen occupy usin therem ainderofthesenotes.

2.1 T he Eulernum berofa �nite dim ensionalvectorbun-

dle

Consider a realvectorbundle � :E ! X overa m anifold X . W e willassum e

thatE and X areorientable,X iscom pactwithoutboundary,and thattherank

(�bredim ension)ofE iseven and satis�esrk(E )= 2m � dim (X )= n.

The Euler class ofE isan integralcohom ology class e(E ) 2 H 2m (X ;R )�

H 2m (X ). Form = 1 (a two-plane bundle)e(E )can e.g.be de�ned in a rather

pedestrian m anner (cf.[21]for the m aterialcovered in this and the �rst part

ofthe following section). W e choose a cover ofX by open sets U� and denote

by g�� :U� \ U� ! SO (2)the transition functionsofE satisfying the cocycle

condition

g�� = g�1

�� ; g��g� = g� : (2.1)

5

Identifying SO (2)� U(1),weset’�� = ilogg�� with

’�� + ’� � ’� 2 2�Z ; (2.2)

so thatd’ isan additivecocycle,

d’�� + d’� = d’� : (2.3)

In fact,m ore than thatistrue.By introducing a partition ofunity subordinate

to fU�g,i.e.a setoffunctions�� satisfying

X

�

�� = 1 ; supp(��)� U� ; (2.4)

and de�ning one-form s�� on U� by �� = (2�)�1P

� d’ � one�ndsthat

1

2�d’�� = �� � �� (2.5)

which obviously im plies (2.3). Thus d�� = d�� on the overlaps U� \ U� and

therefore the d�’spiece togetherto give a globaltwo-form on X which isclosed

butnotnecessarily exact. The cohom ology classofthisform isindependentof

thechoiceof�’ssatisfying (2.5)and istheEulerclasse(E )2 H2(X )ofE .

For higher rank bundles a sim ilar construction is possible in principle but

becom esratherunwieldy.Fortunately thereareother,m oretransparent,waysof

thinking aboute(E ).

The�rstoftheseisin term sofsectionsofE .In general,atwisted bundlewill

have no nowhere-vanishing non-singularsectionsand onede�nestheEulerclass

to bethehom ology classofthezero locusofa genericsection ofE .ItsPoincar�e

dualisthen a cohom ology classin H 2m (X ).

Thesecond m akesuseoftheChern-W eiltheory ofcurvaturesand character-

isticclassesand producesan explicitrepresentativeer (E )ofe(E )in term softhe

curvaturer ofa connection r on E .Thinking ofr asa m atrix oftwo-form s

onehas

er (E )=1

(2�)mPf(r ) (2.6)

wherePf(A)denotesthePfa�an oftherealantisym m etric m atrix A,

Pf(A)=(�1)m

2m m !

X

�a1���a2m A a1a2 :::A a2m �1 a2m ; (2.7)

satisfyingPf(A)2 = det(A).Standard argum entsshow thatthecohom ologyclass

ofer isindependentofthechoiceofr .

Finally,thethird isin term softheThom classofE and wewilldescribethis

in section 2.2.

6

Iftherank ofE isequalto thedim ension ofX (e.g.ifE = TX ,thetangent

bundle ofX )then H 2m (X )= H n(X )= R and nothing islostby considering,

instead ofe(E ),its evaluation on (the fundam entalclass [X ]of)X ,the Euler

num ber

�(E )= e(E )[X ] : (2.8)

In term softhetwodescriptionsofe(E )given above,thisnum bercan beobtained

eitherasthenum berofzerosofa genericsection sofE (which arenow isolated)

counted with m ultiplicity,

�(E )=X

xk:s(xk)= 0

�s(xk) (2.9)

(here�s(xk)isthedegreeorindex ofs atxk),orastheintegral

�(E )=

Z

X

er (E ) : (2.10)

Ofparticularinterest to usis the case where E = TX . The Euler num ber

�(TX )isthen equalto theEulercharacteristic�(X )ofX ,

�(TX )= �(X )�X

k

(�1)kbk(X ) (2.11)

where bk(X ) = dim (H k(X )) is the k’th Bettinum ber ofX . In this context,

equations(2.9)and (2.10),expressing �(X )asthe num berofzerosofa vector

�eld and the integralofa density constructed from the Riem annian curvature

tensorR X ofX ,areknown asthePoincar�e-Hopftheorem and theGauss-Bonnet

theorem respectively. Forexam ple,in two dim ensions(n = 2),(2.10)reducesto

thewellknown form ula

�(X )=1

4�

Z

X

pgd

2xR

whereR isthescalarcurvatureofX .

ForE = TX there is also an interesting generalization of(2.9) involving a

vector �eld V with a zero locus X V which is not necessarily zero-dim ensional.

Denoting theconnected com ponentsofX V by X(k)

V ,thisgeneralization reads

�(X )=X

k

�(X(k)

V ) : (2.12)

Thisreducesto (2.9)when the X(k)

V are isolated pointsand isan identity when

V isthezero vector�eld.

OneofthebeautiesoftheM athai-Quillen form alism ,to bediscussed next,is

thatitprovidesa corresponding generalization of(2.10),i.e.an explicitdi�eren-

tialform representative es;r ofe(E )depending on both a section s ofE and a

connection r on E such that

�(E )=

Z

X

es;r (E ) (2.13)

7

and such that(2.13)reduces to any ofthe above equationsforthe appropriate

choiceofE and s(i.e.to (2.10)ifsisthezero section,to (2.9)when thezerosof

s areisolated,and to (2.12)fora generalvector�eld on TX ).

Ifn > 2m ,then wecannotevaluatee(E )on [X ]asin (2.8).W ecan,however,

evaluate it on hom ology 2m -cycles or (equivalently) take the product ofe(E )

with elem ents ofH n�2m (X )and evaluate thison [X ]. In thisway one obtains

intersection num bersofX associated with thevectorbundleE .A corresponding

interpretation oftheDonaldson polynom ials[12]asobservablesin thetopological

gaugetheory of[1]hasbeen given by Atiyah and Je�rey [11](cf.section 4.2).

2.2 T he T hom class and the M athai-Q uillen form

The Euler class e(E )has the property thatit is the pullback ofa cohom ology

classon E ,called theThom class�(E )ofE ,via thezero section i:X ! E ,

e(E )= i��(E ) : (2.14)

W ewillshow thisexplicitly below (cf.equations(2.33,2.34)).To understand the

origin and signi�canceof�(E ),recallthattherearetwo naturalnotionsofcoho-

m ology fordi�erentialform son a vectorbundle E overa com pactm anifold X :

ordinary deRham cohom ology H �(E )and com pactverticalcohom ology H �

cv(E ).

The latterdealswith form swhose restriction to any �bre hascom pactsupport.

AsE iscontractibleto X onehas

H�(E )’ H

�(X ) : (2.15)

On the other hand,as the com pact cohom ology ofa vector space only has a

generatorin the top dim ensions(a ‘bum p’volum e form with unitvolum e),one

has

H�

cv(E )’ H��2mcv (X ) : (2.16)

M ore technically,for form s ofcom pact verticalsupport one has the notion of

‘push-down’or‘integration alongthe�bres’,denoted by ��.In localcoordinates,

and fortrivialbundles,thisistheobviousoperation ofintegrating overthe�bres

thepartof! 2 �

cv(E )(thespaceofform swith com pactverticalsupport)which

contains a vertical2m -form and interpreting the result as a form on X . This

prescription givesa globally wellde�ned operation

�� :�

cv(E )! ��2m (X ) : (2.17)

In particular,forany ! 2 �

cv(E )and � 2 �(X )onehas

��((���)!)= ���! : (2.18)

�� com m uteswith the exteriorderivativeson E and X (itissu�cientto check

thisin localcoordinates),

��dE = dX �� (2.19)

8

and inducesthe so called Thom isom orphism TE :H �(X )! H �+ 2mcv (E )(2.16).

Underthisisom orphism ,thegenerator12 H 0(X )correspondstoa2m -dim ensional

cohom ology classon E ,theThom class�(E ),

�(E )= T E (1)2 H2mcv (E ) : (2.20)

Byde�nition,�(E )satis�es� ��(E )= 1,sothatby(2.18)theThom isom orphism

isexplicitly given by

TE (�)= (���)�(E ) : (2.21)

After this sm alldigression let us now return to the Euler class e(E ) and

equation (2.14).Asany two sectionsofE arehom otopicasm apsfrom X to E ,

and as hom otopic m aps induce the sam e pullback m ap in cohom ology,we can

use any section s ofE instead ofthe zero section to pullback �(E )to X and

still�nd

s��(E )= e(E ) : (2.22)

The advantage ofthis way oflooking at the Euler class e(E ) should now be

evident: provided that we can �nd an explicit di�erentialform representative

�r (E )of�(E ),depending on a connection r on E ,we can pullitback to X

via a section s to obtain a 2m -form

es;r (E )= s��r (E ) (2.23)

representing theEulerclasse(E )and (ifn = 2m )satisfying (2.13).Itshould be

bornein m ind,however,thatby (2.22)alltheseform sarecohom ologousso that

thisconstruction,asnice asitis,isnotvery interesting from the cohom ological

pointofview. To getsom ething really new one should therefore considersitua-

tionswheretheform s(2.23)arenotnecessarily cohom ologousto er .Aspointed

outby Atiyah and Je�rey [11],such asituation occurswhen oneconsidersin�nite

dim ensionalvectorbundleswhereer (an ‘in�nite-form ’)isnotde�ned atall.In

thatcase the added exibility in the choice ofs becom es crucialand opensup

thepssibility ofobtainingwell-de�ned,buts-dependent,‘Eulerclasses’ofE .W e

willexplain thisin section 2.3.

Toproceed with theconstruction of�r ,letusm aketwoprelim inary rem arks.

The�rstisthatforexplicitform ulaeitisconvenienttoswitch from workingwith

form swith com pactsupportalong the�bresto working with ‘Gaussian shaped’

form rapidlydecreasingalongthe�bres(inasuitabletechnicalsense).Everything

wehavesaid sofargoesthrough in thatsetting[10]and wewillhenceforth replace

�

cv(E )by �

rd(E )etc.

The second isthatPfa�ans(2.7)arise asferm ionic (Berezin)integrals(this

m ay sound likearatherm ysteriousrem ark tom akeatthispoint,butisofcourse

oneofthereasonswhy whatwearegoing through herehasanything to do with

supersym m etry and topological�eld theory). M ore precisely,ifwe have a real

9

antisym m etricm atrix(A ab)and introducerealGrassm ann odd variables�a,then

Pf(A)=

Z

d�e�aA ab�

b=2: (2.24)

In particular,wecan thereforewritetheform er (2.6)as

er (E )= (2�)�mZ

d�e�a

ab

r�b=2: (2.25)

The idea isnow to extend the righthand side of(2.25)to a form �r (E )on E

having Gaussian decay along the�bresand satisfying ���r (E )= 1.

Regarding E asa vector bundle associated to a principalG bundle P with

standard �breF,E = P � G F,we can representform son E by basic,i.e.hori-

zontaland G-invariant,form son P � F,

�(E )= �

bas(P � F) (2.26)

and sectionsofE by G-equivariantm apsfrom P to F.M oreover,via theprojec-

tion� :P ! X ,E pullsbacktothecanonicallytrivialvectorbundle��E = P � F

over P whose induced connection and curvature we also denote by r and r .

W ith thisidenti�cation understood,theThom form � r (E )ofM athaiandQuillen

isgiven by

�r (E )= (2�)�m e�� 2=2

Z

d�e�a

ab

r�b=2+ ir �

a�a (2.27)

wherewehavechosen a�xed �brem etricon F,�a arecoordinateson F and r �a

isthe exteriorcovariantderivative of�a,a one-form on P � F. W e now check

that�r (E )really representstheThom classofE .

Firstofall,integratingout� oneseesthat(2.27)de�nesa2m -form on P � F.

Thisform isindeed basic and represents a closed 2m -form on E . G-invariance

and horizontality arealm ostobviousfrom (2.27)asr and r � arehorizontal(by

the de�nition ofthe covariantexteriorderivative). Lessevidentisthe factthat

�r (E )isclosed.Thisisbestunderstood in term softheequivariantcohom ology

H �

G (F) ofF (cf.sections 5 and 6 of[10]) and is related to the fact that the

exponentin (2.27),

� �2=2+ �a

abr�b=2+ ir �a�a ; (2.28)

isinvariantunderthegraded (i.e.super-)sym m etry

��a = i�a

��a = r �a (2.29)

m apping theGrassm ann odd � to theeven � and � to theGrassm ann odd one-

form r �. ‘On shell’,i.e.using the � equation ofm otion ir �a = abr�b,this

supersym m etry squaresto rotationsby thecurvaturem atrix r ,

�2�a = ab

r�b

�2�a = ab

r�b (2.30)

10

which isthe hallm ark ofequivariantcohom ology. Fora m ore thorough discus-

sion ofthe relation between the classical(Cartan-,W eil-)m odelsofequivariant

cohom ology and theBRST m odel,aswellasoftheM athai-Quillen form alism in

thatcontext,see[23].

By introducing a Grassm ann even scalar�eld B a with ��a = B a and �Ba =

abr�b the‘action’(2.28)becom es�-exacto�-shell,

(2:28)� �(�a(i�a � Ba=2) (2.31)

Itisofcourseno coincidence thatthestructurewehaveuncovered hereisrem i-

niscentoftopological�eld theory,seee.g.(3.11,4.20)below.

Because ofthe factore��2=2,(2.27)iscertainly rapidly decreasing along the

�bre directions. W hat rem ains to be checked to be able to assert that � r (E )

representstheThom class�(E )isthat� ��r (E )= 1 or,undertheisom orphism

(2.26),thatR

F �r (E )= 1.Extracting from the2m -form �r (E )the partwhich

isa 2m -form on F we�nd thatindeed

Z

F

�r (E ) = (2�)�mZ

F

e�� 2=2

Z

d�(id�a�a)

2m

2m !

= (2�)�mZ

F

e�� 2=2

d�1:::d�

2m = 1 : (2.32)

Thisprovesthat

[�r (E )]= �(E )2 H2mrd (E ) : (2.33)

W enow takea closerlook attheform ss��r (E )= es;r (E )(2.23)forvarious

choices ofs. In our notation es;r (E ) is obtained from (2.27) by replacing the

�bre coordinate � by s(x). The �rstthing to note isthatforthe zero section i,

(2.27)reducesto (2.25)and therefore

er (E )= i��r (E ) : (2.34)

Thisisare�nem entof(2.14)toan equality between di�erentialform sand there-

fore,in particular,�nally proves(2.14)itself.

Ifn = 2m and s is a generic section ofE transversalto the zero section,

then we can calculateR

X es;r (E )by replacing s by s for 2 R and evaluating

the integralin the lim it ! 1 .In thatlim itthe curvature term in (2.27)will

notcontributeand onecan usethestationary phaseapproxim ation toreducethe

integraltoasum ofcontributionsfrom thezerosofs,reproducingequation (2.9).

The calculation is entirely analogous to sim ilar calculations in supersym m etric

quantum m echanics(seee.g.[9])and Iwillnotrepeatithere.In fact,aswewill

later derive the M athai-Quillen form ula (2.27) from supersym m etric quantum

m echanics(section 3.3),thisshowsthattherequired m anipulationsarenotonly

entirely analogousto butidenticalwith thosein supersym m etricquantum m ech-

anics.Aswecould equally wellhaveput = 0 in theabove,thisalso establishes

directly theequality of(2.9)and (2.10).

11

Finally,ifE = TX and V isa non-generic section ofX with zero locusX V ,

thesituation isalittlebitm orecom plicated.Itturnsoutthatin thiscaseR

X eV;r

can beexpressed in term softheRiem ann curvaturetensorR X VofX V .HereR X V

arisesfrom the data R X and V entering eV;r via the Gauss-Codazziequations.

Quitegenerally,theseexpressthecurvatureR Y ofasubm anifold Y � X in term s

ofR X and the extrinsic curvature ofY in X (we willrecallthese equations in

section 4.1). Then equation (2.12) is reproduced in the present setting in the

form (weassum ethatX V isconnected -thisisfornotationalsim plicity only)

�(X )=

Z

X

eV;r = (2�)�dim (X V )=2

Z

X V

Pf(R X V) : (2.35)

Again them anipulationsrequired to arriveat(2.35)areexactly asin supersym -

m etricquantum m echanics[17,18]and wewillperform such a calculation in the

contextoftopologicalgaugetheory in section 4.3 (seethecalculationsleading to

(4.31)).

2.3 T he M athai-Q uillen form alism for in�nite dim en-

sionalvector bundles

Let us recapitulate brie y what we have achieved so far. Using the M athai-

Quillen form �r (E ) (2.27),we have constructed a fam ily ofdi�erentialform s

es;r (E )param etrized by a section sand a connection r and allrepresenting the

Eulerclasse(E )2 H 2m (X ). In particular,forE = TX ,the equation �(X )=R

X eV;r (X )interpolates between the classicalPoincar�e-Hopfand Gauss-Bonnet

theorem s.

Tobein asituation wheretheform ses;r arenotnecessarily allcohom ologous

to er ,and where the M athai-Quillen form alism thus ‘com es into its own’[11],

we now consider in�nite dim ensionalvector bundles. To m otivate the concept

ofregularized Euler num ber ofsuch a bundle,to be introduced below,recall

equation (2.12)fortheEulernum ber�(X )ofam anifold X which werepeathere

forconvenience in theform

�(X )= �(X V ) : (2.36)

W hen X is�nite dim ensionalthisisan identity,while itslefthand side isnot

de�ned when X is in�nite dim ensional. Assum e,however,that we can �nd a

vector�eld V on X whose zero locusisa �nite dim ensionalsubm anifold ofX .

Then therighthand sideof(2.36)iswellde�ned and wecan useitto tentatively

de�nea regularized Eulernum ber�V (X )as

�V (X ):= �(X V ) : (2.37)

By (2.13)and thestandard localization argum ents,asre ected e.g.in (2.35),we

expectthisnum berto begiven by the(functional)integral

�V (X )=

Z

X

eV;r (X ) : (2.38)

12

This equation can (form ally) be con�rm ed by explicit calculation. The idea is

again to replace V by V ,so that(2.38)localizesto the zerosofV as ! 1 ,

and to show that in this lim it the surviving term s in (2.28) give rise to the

Riem ann curvaturetensorofX V ,expressed in term sofR X and V viatheGauss-

Codazziequations.A rigorousproofcan probably beobtained in som ecasesby

probabilisticm ethodsasused e.g.by Bism ut[24,25]in related contexts.W ewill,

however,contentourselveswith verifying (2.38)in som eexam plesbelow.

M oregenerally,wearenow led to de�netheregularized Eulernum ber�s(E )

ofan in�nitedim ensionalvectorbundleE as

�s(E ):=

Z

X

es;r (E ) : (2.39)

Again,thisexpression turnsouttom akesensewhen thezero locusofsisa�nite

dim ensionalm anifold X s,in which case�s(E )istheEulernum berofsom e�nite

dim ensionalvector bundle over X s (a quotient bundle ofthe restriction E jX s,

cf.[19,20]).

Ofcourse,there is no reason to expect �s(E )to be independent ofs,even

ifone restricts one’s attention to those sections s for which the integral(2.39)

exists. However,ifs isa section ofE naturally associated with E (we willsee

exam plesofthisbelow),then �s(E )isalso naturally associated with E and can

beexpected to carry interesting topologicalinform ation.Thisisindeed thecase.

Itisprecisely such arepresentation of�nitedim ensionaltopologicalinvariants

byin�nitedim ensionalintegralswhich isthecharacteristicpropertyoftopological

�eld theories. Itisthen perhapsnottoo surprising anym ore atthispoint,that

topological�eld theoryactionscan beconstructed from (2.28)forsuitablechoices

ofX ,E ,and s.

Hereisa survey oftheexam pleswewilldiscussin a littlem oredetailin the

following sections (LM denotes the loop space ofa m anifold M and A k=Gk a

spaceofgaugeorbitsin k dim ensions).

Exam ple 1 X = LM ,E = TX ,V = _x (section 3.2)

(2.28) becom es the standard action SM ofde Rham supersym m etric quantum

m echanicsand Z

LM

eV;r (LM )= Z(SM ) (2.40)

is the partition function ofSM . The zero locus (LM )V ofV is the space of

constantloops,i.e.(LM )V ’ M .W ethereforeexpect(2.40)to calculate

�V (LM )= �(M ) : (2.41)

As this indeed agrees with the wellknown explicit evaluation ofZ(SM ) in the

form

Z(SM )= (2�)�dim (M )=2

Z

M

Pf(R M ) ; (2.42)

13

thisisour�rstcon�rm ation of(2.39).Conversely theM athai-Quillen form alism

now providesan understanding and explanation ofthem echanism by which the

(path)integral(2.40)overLM localizesto theintegral(2.42)overM .

Instead ofthe vector�eld _x one can also use _x + W 0,where W 0 denotesthe

gradientvector�eld ofsom efunction W on M .By an argum enttobeintroduced

in section 3 (the ‘squaring argum ent’) the zero locus ofthis vector �eld is the

zero locusofW 0on M (i.e._x = W 0= 0)whoseEulernum beristhesam easthat

ofM by (2.36),

�V (LM )= �(M W 0)= �(M ) : (2.43)

Again thisagreeswith the explicitevaluation ofthe path integralofthe corre-

sponding supersym m etric quantum m echanicsaction.

Exam ple 2 X = A 4=G4,E = E+ ,s= (FA)+ (section 4.2)

(E+ isa certain bundle ofself-dualtwo-form soverA 4=G4 and (FA)+ isthe self-

dualpartofthecurvatureFA ofA).Thezero locusX s isthem odulispaceM I

ofinstantons,and not unexpectedly the corresponding action is that ofDon-

aldson theory [12,1]. The partition function �s(E+ ) is what is known as the

�rstDonaldson invariantand isonly non-zero when d(M )� dim (M I)= 0. If

d(M )6= 0then onehastoinsertelem entsofH d(M )(A 4=G4)intothepath integral

in them annerexplained attheend ofsection 2.1 to obtain non-vanishing results

(theDonaldson polynom ials).Thisinterpretation ofDonaldson theory isdueto

Atiyah and Je�rey [11].

Exam ple 3 X = A 3=G3,E = TX ,V = �FA (section 4.3)

(� istheHodgeoperator,and theone-form �FA de�nesa vector�eld on A 3=G3,

the gradientvector�eld ofthe Chern-Sim ons functional). The zero locusofV

is the m odulispace M 3 of at connections and the action coincides with that

constructed in [13,14,17].Again one�ndsfullagreem entof

�V (A3=G3)= �(M 3) (2.44)

with the partition function ofthe action which gives�(M 3)in the form (2.35),

i.e.viatheGauss-Codazziequationsfortheem beddingM 3 � A 3=G3.In [11]this

partition function was�rstidenti�ed with a regularized Eulernum berofA 3=G3.

W ehavenow identi�ed itm orespeci�cally with theEulernum berofM 3.In [26]

itwasshown thatforcertain three-m anifolds(hom ology spheres)�V (A3=G3)is

theCasson invariant.HenceourconsiderationssuggestthattheCasson invariant

can bede�ned as�(M 3)form oregeneralthree-m anifolds[17].

Exam ple 4 X = L(A 3=G3),E = TX ,V = _A + �FA (section 4.3)

Thisissupersym m etricquantum m echanicson A 3=G3 and in a sensea com bina-

tion ofallthethreeaboveexam ples.Theresulting (non-covariant)gaugetheory

action in 3+ 1 dim ensionsisthatofDonaldson theory (exam ple2).Afterpartial

localization from L(A 3=G3) to A 3=G3 it is seen to be equivalent to the action

ofexam ple 3. Further reduction to the zeros ofthe gradient vector �eld �FA(exam ple1)reducesthepartition function to an integraloverM 3 and calculates

14

�(M 3).Thisagain con�rm stheequalityoftheleftand righthand sidesof(2.38).

The reason why Donaldson theory is related to instanton m odulispaces in ex-

am ple2,butto m odulispacesof atconnectionsin thisexam pleisexplained in

[17].

3 T he Euler N um ber of Loop Space and Su-

persym m etric Q uantum M echanics

In this section we willwork out som e ofthe details ofexam ple 1. W e begin

with a (very)briefsurvey ofthe geom etry ofloop space (section 3.1). W e then

apply the M athai-Quillen form alism to the tangentbundle ofloop space,derive

supersym m etric quantum m echanics from that,and review som e ofthe m ost

im portant features ofsupersym m etric quantum m echanics in the light ofthis

derivation (section 3.2). Finally,to com plete the picture,we explain how the

�nite-dim ensionalM athai-Quillen form (2.27)can bederived from supersym m et-

ricquantum m echanics(section 3.3).

3.1 Loop space geom etry

W e denote by M a sm ooth orientable Riem annian m anifold with m etric g and

by LM theloop spaceofM ,i.e.thespaceofsm ooth m apsfrom thecircleS1 to

M ,

LM := C1 (S1

;M ) (3.1)

(consistentwith thesloppynesstobeencountered throughoutthesenoteswewill

notworry aboutthe technicalities ofin�nite dim ensionalm anifolds). Elem ents

ofLM are denoted by x(t)orx�(t),where t2 [0;1],x� are (local)coordinates

on M and x�(0)= x�(1).In supersym m etricquantum m echanicsitisconvenient

to scale tsuch thatt2 [0;�]and x�(0)= x�(�)forsom e � 2 R ,and to regard

� asan additionalparam eter(theinverse tem perature)ofthetheory.

A tangentvectortoaloop x(t)can beregarded asan in�nitesim alvariation of

theloop.Assuch itcan bethoughtofasa vector�eld on theim agex(S1)� M

(tangenttoM butnotnecessarily totheloop x(S1)).In otherwords,thetangent

space Tx(LM ) to LM at the loop x(t) is the space ofsm ooth sections ofthe

tangentbundleTM restricted to theloop x(t),

Tx(LM )’ �1 (x�(TM )) : (3.2)

There isa canonicalvector�eld on LM which generatesrigid rotationsx(t)!

x(t+ �) ofthe loop around itself. Itisgiven by V (x)(t)= _x(t)(orV = _x for

short).Them etricg on M inducesa m etric g on LM through

gx(V1;V2)=

Z1

0

dtg��(x(t))V�

1 (x)(t)V�2 (x)(t) : (3.3)

15

Likewise,every p-form � on M givesriseto a p-form � on LM via

�x(V1;:::;Vp)=

Z1

0

dt�x(t)(V1(x)(t);:::;Vp(x)(t)) ; (3.4)

and a localbasisofone-form son LM isgiven by thedi�erentialsdx�(t).

The lastpiece ofinform ation we need isthatthe Levi-Civit�a connection on

M can be pulled back to S1 via a loop x(t). Thisde�nesa covariantderivative

on (3.2)and itsdualwhich wedenoteby r t.W ehavee.g.

(r tV�)(x)(t)= d

dtV�(x)(t)+ ����(x(t))_x

�(t)V �(x)(t) : (3.5)

3.2 Supersym m etric quantum m echanics

W e are now in a position to discuss exam ple 1 ofsection 2.3 in m ore detail.

In the notation ofthat section,we choose X = LM ,E = TX ,and V = _x.

Theanticom m uting variables�a thusparam etrizethe�bresofTX and wewrite

them as�a = e�a� � wheree

�a istheinversevielbein corresponding to g��.Using

the m etric (3.3)asa �bre m etric on TxX ,the �rstterm of(2.28)issim ply the

standard bosonickineticterm ofquantum m echanics,

�2=2!

Z �

0

dtg�� _x� _x�=2 : (3.6)

To puttherem aining term sinto a m orefam iliarform ,weusethestandard trick

ofreplacing thedi�erentialsdx�(t)by periodicanticom m uting variables,

dx�(t)!

�(t) (3.7)

and integrating overthem aswell. Asthe integraloverthe ’swillsim ply pick

outthetop-form partwhich isthen tobeintegrated overX (cf.(2.39)),nothingis

changed bythesubstitution (3.7).W ith allthisin m ind thecom pleteexponential

oftheM athai-Quillen form eV;r (LM )becom es

SM =

Z �

0

dt[�g�� _x� _x�=2+ R

����� �

� � � �=4� i� �r t

�] : (3.8)

This is precisely the standard action ofde Rham (or N = 1) supersym m etric

quantum m echanics to be found e.g.in [27,28,29,30,31](with the spinors

appearing there decom posed into theircom ponents;we also choose and � to

be independentreal�eldsinstead ofcom plex conjugates). Itwillbe convenient

to introduce a m ultiplier �eld B � and to rewrite the action (3.8)in �rst order

form ,

SM =

Z �

0

dt[i_x�B � + g��B �B �=2+ R

����� �

� � � �=4� i� �r t

�] : (3.9)

16

Thesupersym m etry ofthisaction is

�x� =

�; �� � = B � � ����

� � �

� � = 0 ; �B� = ����B �

� � R����

� � � �=2 : (3.10)

Thisisreadily veri�ed by noticingthat�2 = 0and that(3.9)can itselfbewritten

asa supersym m etry variation,

SM = �

Z �

0

dt[� �(i_x� + g

��B �=2)] : (3.11)

Note the sim ilarity with (2.31). Reinterpreting � asa BRST operator,thisalso

showsthatthesectorofsupersym m etric quantum m echanicsannihilated by � is

topological(a BRST exactaction being oneofthehallm arksoftopological�eld

theory).Aswewillseebelow thatonly groundstatescontributeto thepartition

function anyway,itis,in particular,independentofthecoe�cientofthesecond

term of(3.11)regardlessofwhetherwetreat� asa conventionalsupersym m etry

operator (m apping bosonic to ferm ionic states and vice-versa) or as a BRST

operator(annihilating physicalstates).Rescaling thisterm by a realparam eter

� we�nd theequivalentaction

SM =

Z �

0

dt[�g�� _x� _x�=2� + �R

����� �

� � � �=4� i� �r t

�] : (3.12)

On the other hand,ifwe rescale the tim e variable by � we obtain the action

(3.12) withR�

0dt replaced by

R1

0dt and � replaced by �. Thus the ‘topolog-

ical’�-independence translates into the quantum m echanical�-independence.

Conversely,this�-independence isobviousfrom the standard Ham iltonian con-

struction ofsupersym m etric quantum m echanics(cf.below)and translatesinto

thetopological�-independenceof(3.12).

This is not the place to enter into a detailed discussion ofsupersym m etric

quantum m echanics,and wewillin thefollowing focuson thoseaspectsrelevant

for the M athai-Quillen side ofthe issue and our subsequent considerations in-

volving topologicalgauge theories. For detailed discussions ofsupersym m etric

quantum m echanics in the contextofindex theory and topological�eld theory

thereaderisreferred to [30]and [9,pp.140-176]respectively.

Our discussion ofthe M athai-Quillen form alism suggests that the partition

function Z(SM )ofthesupersym m etricquantum m echanicsaction SM (3.8),with

periodic boundary conditionson allthe �elds,isthe Eulernum ber�(M )ofM

(as �V (LM ) = �((LM )V ) = �(M ),cf.(2.37-2.41)). As is wellknown,this is

indeed thecase.

Theconventionalway to seethis(ifonedoesnotyettrustthein�nitedim en-

sionalversion oftheM athai-Quillen form alism )isto startwith thede�nition of

�(M )astheEulercharacteristicofM (2.11).Asthereisaone-to-onecorrespon-

dence between cohom ology classes and harm onic form s on M (m ore precisely,

17

thereisa uniquerepresentative in every deRham cohom ology classwhich isan-

nihilated by the Laplacian � = dd � + d�d)one can write �(X )asa trace over

thespaceKer�,

�(M )= trK er� (�1)F; (3.13)

where(�1)F is+1 (�1)on even (odd)form s.Astheoperatord+ d� com m utes

with � and m aps even to odd form s and vice-versa,there is an exact pairing

between ‘bosonic’and ‘ferm ionic’eigenvectorsof� with non-zero eigenvalue.It

is thus possible to extend the trace in (3.13) to a trace over the space ofall

di�erentialform s,

�(M )= tr �(�1)F e��� : (3.14)

As only the zero m odes of� willcontribute to the trace,it is evidently inde-

pendentofthevalueof�.Onceonehasput�(M )into thisform ofa statistical

m echanicspartition function,onecan usetheFeynm an-Kacform ula torepresent

itasa supersym m etric path integral[30]with the action (3.8),im aginary tim e

ofperiod � and periodicboundary conditionson theanticom m uting variables �

(dueto theinsertion of(�1)F ).Conversely,a Ham iltonian analysisoftheaction

(3.8) would tellus that we can represent its Ham iltonian by the Laplacian �

on di�erentialform s[28]and,tracing back the stepswhich led usto (3.14),we

would then again deducethatZ(SM )= �(M ),asanticipated in (2.40).

ThisHam iltonian way ofarriving atthe action ofsupersym m etric quantum

m echanicsshould becontrasted with theM athai-Quillen approach.In theform er

one startswith the operatorwhose index one wishes to calculate (e.g.d + d�),

constructs a corresponding Ham iltonian,and then deduces the action. On the

otherhand,in thelatteronebeginswith a�nitedim ensionaltopologicalinvariant

(e.g.�(M ))and representsthatdirectly asan in�nite dim ensionalintegral,the

partition function ofa supersym m etric action.

W hatm akessuch a path integralrepresentation of�(M )interesting isthat

one can now go ahead and try to som ehow evaluate it directly,thus possibly

obtaining alternativeexpressionsfor�(M ).Indeed,onecan obtain path integral

‘proofs’ofthe Gauss-Bonnet and Poincar�e-Hopftheorem s in this way. This is

justthe in�nite dim ensionalanalogue ofthe considerationsofsection 2.2 where

di�erent choices ofs inR

X es;r (E ) lead to di�erent expressions for �(E ). As

we willderive the �nite dim ensionalM athai-Quillen form from supersym m etric

quantum m echanicsin section 3.3 wecan appealto them anipulationsofsection

2.2 to com plete these ‘proofs’. However,itis also instructive to perform these

calculations directly. Before indicating how this can be done,we willneed to

introduce a generalization ofthe action (3.8) which arises when one takes the

section _x� + g��@�W ofT(LM )(cf.exam ple 1 ofsection 2.3)to regularize the

Eulernum berofLM . Here W isa function (potential)on M and isyetone

m ore arbitrary realparam eter. In that case one obtains (introducing also the

18

param eter� of(3.12))

SM ; W =

Z�

0

dt[i(_x� + g��@�W (x))B � + �g

��B �B �=2+ �R

����� �

� � � �=4

�i� �(���r t+ g

��r �@�W ) �] : (3.15)

From theHam iltonian pointofview thisaction arisesfrom replacing theexterior

derivatived by

d ! d W � e� W

de W

: (3.16)

and applyingtheaboveproceduretothecorrespondingLaplacian � W .Asthere

isa one-to-one correspondence between �-and � W -harm onic form s,thisalso

represents�(M )(independently ofthevalueof ).

Thisfreedom in the choice ofparam eters�;�; greatly facilitatesthe eval-

uation ofthe partition function. Let us,for exam ple,choose � = 0 in (3.15).

Then the curvature term drops out com pletely and the B -integralwillsim ply

give usa delta function constraint _x� + g��@�W = 0. Squaring thisequation

and integrating itovertone�nds

_x� + g��@�W = 0

!

Z �

0

dtg�� _x� _x� +

2g��@�W @�W + 2 _x�@�W = 0

! _x� = 0= @�W (3.17)

as the second line is the sum oftwo nonnegative term s and a totalderivative.

Thisisthe ‘squaring argum ent’referred to in section 2.3. Itdem onstratesthat

the path integralover LM is reduced to an integralover M (by _x� = 0) and

furtherto an integraloverthesetM W 0 ofcriticalpointsofW (and analogously

forthe ’sby supersym m etry).W hen thecriticalpointsareisolated,inspection

of(3.15)im m ediately revealsthatthepartition function is

�(M )= Z(SM ;W )=X

xk:dW (xk= 0

sign(detH xk(W )) ; (3.18)

where

H xk(W )= (r �@�W )(xk) (3.19)

istheHessian ofW atxk.ThisisthePoincar�e-Hopftheorem (2.9).Thisresult

can also be derived by keeping � non-zero and taking the lim it ! 1 instead

which also hasthee�ectoflocalizing thepath integralaround thecriticalpoints

ofW becauseoftheterm 2W 02 in theaction.

Ifweswitch o� thepotential,then we can notsim ply set� = 0 in (3.15),as

the resulting path integralwould be singulardue to the undam ped bosonic and

ferm ionic zero m odes. In thatcase,the lim it� ! 0 or� ! 0 hasto be taken

with m orecare.Sincewhateverwecan dowith � wecan alsodowith � letusset

19

� = 1 in thefollowing.W e�rstrescale thetim ecoordinatetby �,and then we

rescaleB and � by �1=2,B ! �1=2B and � ! �1=2 � ,and allthenon-zero-m odes

ofx and by ��1=2 . This willleave the path integralm easure invariant and

hasthe e�ectthatallthe �-dependentterm sin the action are atleastoforder

O (�1=2)and thelim it� ! 0 can now betaken with im punity.Theintegralover

thenon-constantm odesgives1 and thenet-e�ectofthisisthatoneisleftwith

a �nite-dim ensionalintegraloftheform (2.25),nam ely

�(M )�

Z

dx

Z

d

Z

d� eR��

��� �

� � � � =4

; (3.20)

overtheconstantm odesofx, ,and � ofwhich therearedim (M )each.In order

to get a non-zero contribution (i.e. to soak up the ferm ionic zero m odes) one

hasto expand (3.20)to (dim (M )=2)’th order,yielding the Pfa�an ofR M and

hence,upon integration overM (the x zero m odes)the Gauss-Bonnettheorem

(2.6,2.10). (3.20) also gives the correct result for odd dim ensionalm anifolds,

�(M )= 0,asthere isno way to pulldown an odd num berof ’sand � ’sfrom

theexponent.

Ifthe criticalpoints ofW are not isolated then,by a com bination ofthe

aboveargum ents,onerecoversthegeneralization �(M )= �(M W 0)(2.12,2.43)of

thePoincar�e-Hopftheorem in theform (2.35).

Asthistreatm entofsupersym m etricquantum m echanicshasadm ittedlybeen

som ewhatsketchy Ishould perhaps,sum m arizing thissection,stateclearly what

aretheim portantpointsto keep in m ind:

1.TheM athai-Quillen form alism applied to theloop spaceLM ofa Riem an-

nian m anifold M leadsdirectly to the action ofsupersym m etric quantum

m echanicswith targetspaceM .Di�erentsectionslead to di�erentactions,

and thosewehaveconsidered allregularizetheEulernum berofLM to be

�(M ).

2.Explicit evaluation ofthe supersym m etric quantum m echanics path inte-

gralsobtained in thisway con�rm sthatwe can indeed representthe reg-

ularized Euler num ber �V (LM ),as de�ned by (2.37),by the functional

integral(2.38).

3.Finally,Ihaveargued (although notproved in detail)thatthezero m odes

are allthat m atter in supersym m etric quantum m echanics, the integral

overthe non-zero-m odesgiving 1.Thisobservation isusefulwhen one at-

tem ptsto constructtopologicalgaugetheoriesfrom supersym m etricquan-

tum m echanicson spacesofconnections(see[18]and therem arksin section

4.3).

20

3.3 T he M athai-Q uillen form from supersym m etricquan-

tum m echanics

So farwehavederived theaction ofsupersym m etric quantum m echanicsby for-

m ally applyingtheM athai-Quillen form alism toLM ,and wehaveindicated how

torederivetheclassical(generalized)Poincar�e-Hopfand Gauss-Bonnetform ulae.

W hatisstilllacking to com pletethepictureisa derivation ofthegeneral(�nite

dim ensional) M athai-Quillen form �r (E ) (2.27) for E = TM from supersym -

m etricquantum m echanics.

As�r (TM )can bepulled back to M via an arbitrary vector�eld (section of

TM )v,notnecessarily agradientvector�eld,weneed toconsiderthesupersym -

m etricquantum m echanicsaction resulting from theregularizing section _x + v

ofT(LM ). Thisisjustthe action (3.15)with @�W replaced by g��v�. In that

case the squaring argum ent,asexpressed in (3.17),failsbecause the cross-term

willnotintegrateto zero.In thelim it ! 1 thepath integralwillnevertheless

reduce to a Gaussian around the zero locusofv because ofthe term 2g��v�v�

in theaction,and in thislim itthepath integralcalculates�(M )= �(M v)in the

form (2.35).

To derivetheM athai-Quillen form ,however,weareinterested in �nitevalues

of . Thus,what we need to do now is adjust the param eters in such a way

thatthezero m odesofalltheterm sinvolving thevector�eld v orthecurvature

survive. Proceeding exactly as in the derivation ofthe Gauss-Bonnet theorem

oneendsup with a tim e-independent‘action’oftheform

B2=2+ v

�B � + R

����� � � �

� �=4� i � �r �v

� � (3.21)

which -upon integration over B -reproduces precisely the exponent (2.28) of

theM athai-Quillen form (2.27)with �a replaced by thearbitrary section v� or

ea�v� ofTM . W e have thus also rederived the M athai-Quillen form ula (2.13)

forTM ,

�(M )=

Z

M

ev;r (TM ) ; (3.22)

from supersym m etric quantum m echanics. Specializing now to v = 0 or v a

genericvector�eld with isolated zerosagain reproducestheclassicalexpressions.

4 T he Euler N um ber of Vector B undles over

A =G and TopologicalG auge T heory

In thissection weessentially work outthedetailsofexam ples2and 3and discuss

som e related m odelsaswell. Section 4.1 containsa briefsum m ary ofthe facts

wewillneed from thegeom etry ofgaugetheories.In section 4.2 wewillseehow

Donaldson theory can be interpreted in term softhe M athai-Quillen form alism .

Section 4.3 sketches the construction ofa topologicalgauge theory in 3d from

21

the tangent bundle over or(alternatively) supersym m etric quantum m echanics

on gauge orbit space which represents the Euler characteristic ofthe m oduli

spaceof atconnections.Italso containsa briefdiscussion ofthe2d analogueof

Donaldson theory.

4.1 G eom etry ofgauge theories

Let(M ;g)bea com pact,oriented,Riem annian m anifold,� :P ! M a principal

G bundleoverM ,G a com pactsem isim ple Liegroup and g itsLiealgebra.W e

denote by A the space of(irreducible)connectionson P,and by G the in�nite

dim ensionalgauge group ofverticalautom orphism sofP (m odulo the centerof

G).Then G actsfreely on A and

�:A ! A =G (4.1)

is a principalG bundle. The aim ofthis section willbe to determ ine a con-

nection and curvature on this principalbundle,so thatwe can write down (or

recognize)theM athai-Quillen form forsom ein�nitedim ensionalvectorbundles

associated to it. W e willalso state the Gauss-Codazziequation which express

theRiem ann curvaturetensorR M ofsom em odulisubspaceM ofA =G in term s

ofthe curvature ofA =G and the extrinsic curvature (second fundam entalform )

oftheem bedding M ,! A =G.Thedetailscan befound e.g.in [32,33,34,35].

Continuing with notation,wedenoteby k(M ;g)thespaceofk-form son M

with valuesin theadjointbundleadP := P � ad g and by

dA :k(M ;g)! k+ 1(M ;g) (4.2)

thecovariantexteriorderivativewith curvature(dA)2 = FA.Thespaces

k(M ;g)

havenaturalscalarproductsde�ned bythem etricgon M (and thecorresponding

Hodgeoperator�)and an invariantscalarproducttr on g,nam ely

hX ;Y i=

Z

M

tr(X � Y ) ; X ;Y 2 k(M ;g) (4.3)

(Ihopethatoccasionally denoting these form sby X aswellwillnotgiveriseto

any confusion with the m anifold X ofsection 2). The tangentspace TAA to A

ata connection A can be identi�ed with 1(M ;g)(asA isan a�ne space,two

connectionsdi�ering by an elem entof 1(M ;g)). Equation (4.3)thusde�nesa

m etric gA on A .The Lie algebra ofG can be identi�ed with 0(M ;g)and acts

on A 2 A via gaugetransform ations,

A 7! A + dA� ; �2 0(M ;g) ; (4.4)

so thatdA� is the fundam entalvector �eld atA corresponding to �. At each

pointA 2 A ,TAA can thusbe splitinto a verticalpartVA = Im (dA)(tangent

22

to the orbitofG through A)and a horizontalpartH A = Ker(d�A)(theorthogo-

nalcom plem entofVA with respectto the scalarproduct(4.3)). Explicitly this

decom position ofX 2 1(M ;g)into itsverticaland horizontalpartsis

X = dAG0

Ad�

AX + (X � dAG0

Ad�

AX ) ;

� vAX + hAX ; (4.5)

whereG 0A = (d�AdA)

�1 istheGreensfunction ofthescalarLaplacian (which exists

ifA isirreducible).W ewillidentify thetangentspaceT[A ]A =G with H A forsom e

representative A ofthegaugeequivalenceclass[A].

Then gA inducesa m etricgA =G on A =G via

gA =G([X ];[Y ])= gA (hAX ;hAY ) ; (4.6)

where X ;Y 2 1(N ;g)projectto [X ];[Y ]2 T[A ]A =G. W ith the sam e notation

theRiem annian curvatureofA =G is

hR A =G([X ];[Y ])[Z];[W ]i = h�[hAX ;�hAW ];G 0

A � [hAY;�hAZ]i� (X $ Y )

+ 2h�[hAW ;�hAZ];G0

A � [hAX ;�hAY ]i : (4.7)

IfM issom eem bedded subm anifold ofA =G,then (4.6)inducesa m etricgM on

M whoseRiem ann curvaturetensoris

hR M ([X ];[Y ])[Z];[W ]i = hR A =G([X ];[Y ])[Z];[W ]i

+ (hK M ([Y ];[Z]);K M ([X ];[W ])i� (X $ Y )) ;(4.8)

whereK M istheextrinsiccurvature(orsecond fundam entalform )ofM in A =G.

Forinstanton m odulispacesK M hasbeen com puted in [35]and form odulispaces

of atconnectionsin two and threedim ensionsone�nds[17]

K M ([X ];[Y ])= �d�AG2

A[�X A;�YA] : (4.9)

Here the tangent vectors [X ]and [Y ]to M are represented on the right hand

sideby elem ents �X and �Y of1(M ;g)satisfying both thehorizontality condition

d�A�X = d�A

�Y = 0 and thelinearized atnessequation dA �X = dA �Y = 0.G 2A isthe

Greensfunction oftheLaplacian on two-form sand in thethree-dim ensionalcase

wethinkofitasbeingcom posed withaprojectorontotheorthogonalcom plem ent

ofthezero m odesoftheLaplacian.Thus

hK M ([Y ];[Z]);K M ([X ];[W ])i= h[�YA;�ZA];G2

A[�X A;�W A]i (4.10)

and togetherwith (4.7)and (4.8)thisdeterm inesR M entirely in term sofGreens

functionsofdi�erentialoperatorson M .Itisin thisform thatwewillencounter

R M in section 4.3.

23

The decom position (4.5) also de�nes a connection on the principalbundle

A ! A =G itself,with connection form �A = G 0Ad

�

A. Indeed,�A can be regarded

asa Liealgebra (= 0(M ;g))valued one-form on A ,

�A :TAA ! 0(M ;g)

X 7! �A (X )= G0

Ad�

AX : (4.11)

It transform s hom ogenously under gauge transform ations,is obviously vertical

(i.e.vanisheson Ker(d�A)),and assignsto the fundam entalvector�eld dA� the

corresponding Liealgebra elem ent

�A (dA�)= G0

Ad�

AdA�= � ; (4.12)

asbehovesa connection form .Itscurvatureisthehorizontaltwo-form

� A = dA �A +1

2[�A ;�A ] (4.13)

(dA denotestheexteriorderivativeonA ).Evaluated onhorizontalvectorsX ;Y 2

H A thesecond term iszero and from the�rstterm only thevariation ofA in d�Awillcontribute (because otherwise the surviving d�A willannihilate either X or

Y ).Thusone�nds

� A (X ;Y )= G0

A � [X ;�Y ] ; (4.14)

a form ula thatwewillreencounterin ourdiscussion ofDonaldson theory below.

Finally,we willintroduce the bundles E0 and E+ which willplay a role in

theinterpretation oftopologicalgaugetheoriesfrom theM athai-Quillen pointof

view below.Ifdim (M )= 2,weconsiderthebundle

E0 := A � G 0(M ;g) (4.15)

associatedtotheprincipalbundle(4.1)viatheadjointrepresentation.Ifdim (M )=

4,we choose as�bre the space 2+ (M ;g)ofself-dualtwo-form s. One then has

theassociated vectorbundle

E+ := A � G 2

+(M ;g) (4.16)

overA =G.In thestandard m anner(4.15)and (4.16)inherittheconnection (4.11)

and itscurvature(4.14)from theparentprincipalbundleA ! A =G (4.1).

4.2 T he A tiyah-Je�rey Interpretation ofD onaldson the-

ory

Donaldson theory [1]is the prim e exam ple ofa cohom ological�eld theory. It

wasintroduced by W itten to givea �eld theoreticdescription oftheintersection

24

num bersofm odulispacesofinstantonsinvestigated by Donaldson [12].Donald-

son’s introduction ofgauge theoretic m ethods into the study offour-m anifolds

hashad enorm ousim pacton thesubject(see[36]forreviews),butunfortunately

it would require a seperate set oflectures to describe at least the basic ideas.

Likewise,itis notpossible to give an account ofthe �eld theoretic description

here which would do justice to the m any things that can and should be said

aboutDonaldson theory. Therefore,Iwillm ake only a few generalrem arkson

the structure ofthe action ofDonaldson theory and other cohom ological�eld

theoriesdescribing intersection theory on m odulispaces. The m ain aim ofthis

section will,ofcourse,beto show thatthisaction is,despiteappearance,also of

theM athai-Quillen type.Forareview ofboth them athem aticaland thephysical

sideofthestory see[9,pp.198-247].

The action ofDonaldson theory on a four-m anifold M in equivariant form

(i.e.priorto theintroduction ofgaugeghosts)is[1]

SD =

Z

M

�

B + (FA)+ + �+ (dA )+ � �B2

+=2+ �dA �

�

+�

��dA � dA� + ��[ ;� ]� ��[�+ ;�+ ]=2�

: (4.17)

Here(:)+ denotesprojection onto theself-dualpartofa two-form ,

(FA)+ =1

2(FA + �FA) ; �(FA)+ = (FA)+ ; (4.18)

etc.Furtherm ore 2 1(M ;g)isa Grassm ann odd Liealgebra valued one-form

with ghostnum ber1. Itis(asin supersym m etric quantum m echanics) the su-

perpartnerofthefundam entalbosonicvariableA and representstangentvectors

to A .(B + ;�+ )areself-dualtwo-form swith ghostnum bers(0;�1)(Grassm ann

parity (even,odd)),and (�;��;�) are elem ents of0(M ;g) with ghost num bers

(2;�2;�1)and parity(even,even,odd).� isarealparam eterwhosesigni�canceis

thesam easthatplayed by � in supersym m etricquantum m echanics(cf.(3.15)).

Thisaction hasan equivariantly nilpotentBRST-likesym m etry

�A = � = �dA�

��+ = B + �B+ = [�;�+ ]

��� = � �� = [�;��]

�� = 0 �2 = �� (4.19)

where�� denotesagaugevariation with respectto�.From thesetransform ations

itcan beseen thattheaction SD isBRST-exact,

SD = �

Z

M

�+ ((FA)+ � �B + =2)+ ��dA � : (4.20)

(cf.(2.31,3.11)).Thesinglem ostim portantconsequenceof(4.20),which wewill

abbreviate to SD = ��D ,is that the partition function Z(SD ) ofSD is given

25

exactly by itsone-loop approxim ation.Likewise,itisindependentofthem etric

on M and any other‘couplingconstants’which m ay enterintoitsconstruction in

addition to�h and g��.E.g.forthem etrictheargum entrunsasfollows.Although

g�� entersin a num berofplacesin (4.17),a variation ofitproducesan insertion

ofaBRST-exactoperatorintothepath integralwhosevacuum expectation value

vanishesprovided thatthevacuum isBRST invariant,

�

�g��Z(SD ) = �

�g��

Z

e��� D

= �h0j�( �

�g���D )j0i= 0 : (4.21)

By the sam e argum ent,Z(SD )isindependentof� and correlation functionsof

m etric independent and BRST invariant operators are them selves m etric inde-

pendent. W e willbrie y com e back to these ‘observables’ofDonaldson theory

below.

Equation (4.20)also m akesthe signi�cance ofthe individualterm sin (4.17)

m ore transparent. In particular,one sees thatthe �rst term of(4.20)im poses

a delta function (� = 0)orGaussian (for� 6= 0)constraintonto the instanton

con�gurations(FA)+ = 0.Togetherwith the gauge�xing ofthe gauge �eldsA,

im plicitin theabove,thislocalizesthepath integralaround theinstanton m oduli

spaceM I.Thesecond term ,on theotherhand,�xesthetangentvector to be

horizontal,i.e.tosatisfyd�A = 0,and thusrepresentsatangentvectortoA =G.

M oreover,the �+ equation ofm otion restricts furtherto be tangentto M I,

i.e.to satisfy the linearized instanton equation (dA )+ = 0 (m odulo irrelevant

term sproportionalto �).Thenum berof zero m odeswillthus(generically,see

[1,9])beequalto thedim ension d(M )ofM I.

The structure ofDonaldson theory sum m arized in the preceding paragraphs

isprototypicalforthe actionsofcohom ological�eld theories in general: Given

them odulispace M ofinterest,oneseeksa description ofitin term sofcertain

�elds(e.g.connections),�eld equations(e.g.(F A )+ = 0),and theirsym m etries

(e.g.gauge sym m etries). One then constructs an action which is essentially a

bunch ofdelta functions or Gaussians around the desired �eld con�gurations

and (by supersym m etry) their tangents. Thus,a topologicalaction describing

intersection theory on the m odulispace of atconnectionson som e n-m anifold

M would roughly beoftheform

S �

Z

M

B n�2 FA + (super partners)+ (gauge �xing term s) ; (4.22)

whereB 2 n�2 (M ;g)and (forthecognoscenti)‘gauge�xing term s’ism eantto

also include alltheterm scorresponding to thehighercohom ology groupsofthe

deform ation com plex ofM ,i.e.to the tower ofBianchisym m etries �B B n�2 =

dAB n�3 ;�B B n�3 = :::.

Evidently,thisisquitea pragm aticand notvery sophisticated way oflooking

attopological�eld theory. Itwill,however,be good enough forthe tim e being.

26

Lateron wewillseehow toconstructtheaction (4.22)from them oresatisfactory

M athai-Quillen point ofview. For an elaboration ofthe axiom atic approach

initiated by Atiyah [37]see[38,chs.3 and 4].

Letusnow return to Donaldson theory and show thatitsaction SD isofthe

M athai-Quillen form .W ewilldo thisby m aking useoftheequationsofm otion

arising from (4.20)(which islegitim atesincealltheintegralsareGaussian).W e

set� = 1 in thefollowing.

� Integrating outB oneobtainstheterm �(FA)2+ =2

� The�-equation im pliesthat ishorizontalwhich ishenceforth tacitly un-

derstood

� The�� equation ofm otion yields

� = G0

A � [ ;� ] ; (4.23)

and,plugged back into theaction,thisgivesriseto theterm

�[�+ ;�+ ]G0

A[ ;� ]=2 :

� Putting allthis together we see that e�ectively the action ofDonaldson

theory is

SD =

Z

M

�(FA )2

+=2� [�+ ;�+ ]G

0

A[ ;� ]=2+ (dA )+ �+ : (4.24)

Let us now com pare this with (2.28). W e see that,apart from a factor ofi

which isnotterribly im portantand which can besm uggled back into (4.17)and

(4.24)by appropriate scaling ofthe �elds),the correspondence isperfect.From

the identi�cation �a � �+ we read o� that the standard �bre ofthe sought

forvector bundle is 2+(M ;g). The section is obviously s(A)= (FA )+ ,and as

thistransform sin theadjointundergaugetransform ationsthevectorbundle in

question hasto bethebundleE+ introduced in (4.16).Thisisalso con�rm ed by

a com parison ofthesecond term of(2.28)with thesecond term of(4.24)and the

curvatureform � A (4.14).Thuswe�nally arriveatthedesired equation [11]

Z(SD )= �s(E+ ) (4.25)

identifying the partition function ofDonaldson theory as the regularized Eu-

lernum ber ofthe in�nite dim ensionalvectorbundle E+ and proving the result

claim ed in exam ple 2 ofsection 2.3.

One im portant point we have ignored so far is that the partition function

Z(SD )willbe zero whenever there are zero m odes,i.e.whenever the dim en-

sion d(M ) ofM I is non-zero. This is in m arked contrast with the situation

27

we encountered in supersym m etric quantum m echanics in section 3. There the

partition function Z(SM )= �(M )wasgenerally non-zero,despite the presence

ofdim (M ) zero m odes. Iwillnow brie y try to explain the reason for this

di�erenceand therelated issueofobservablesin Donaldson theory(with noclaim

to com pletenessnorto com pletecom prehensibility):

In supersym m etric quantum m echanicsthere are an equalnum berof and� zero m odes,and these can be soaked up by expanding the curvature term

(which containsan equalnum berof ’sand � ’s)to the appropriate power. In

Donaldson theory theroleof � isplayed by �+ .Generically,however,therewill

beno �+ zero m odesatall,independently ofthedim ension ofthem odulispace,

so thattheferm ionic zero m odescan notbesoaked up by thecurvatureterm

of(4.24). (As an aside: the �+ zero m odes represent the second cohom ology

group ofthe instanton deform ation com plex and thus,together with reducible

connections,the obstruction to having a sm ooth m odulispace. Forthe classof

four-m anifoldsconsidered in [12]itcan beshown thatthiscohom ology group is

zero atirreducibleinstantonsfora genericm etric.)

Thus,in orderto geta non-zero result one has to insert operatorsinto the

path integralwhich take care ofthe zero m odes or,in other words,one has

to construct a top-form on M I which can then be integrated over it. These

operatorshavetobeBRST invariant,and -in view of(4.19)-thistranslatesinto

therequirem entthatthey representcohom ology classesofA =G.Thisisjustlike

the situation we considered atthe end ofsection 2. W hen there isa m ism atch

between the rank 2m ofE and the dim ension n ofX one can obtain non-zero

num bersby pairing er (E )with representatives ofH n�2m (X ). Likewise,even if

n = 2m but one chooses a non-generic section ofE with a k-dim ensionalzero

locus,thiscan berepresented by an (n� k)-form which stillhastobepaired with

ak-form in ordertom akeitavolum eform on X .In thecaseofDonaldson theory

wehavechosen asection with ad(M )-dim ensionalzerolocusand wehavetopair

thecorresponding Eulerclass,theintegrand of(4.25),with d(M )-form son A =G

to produce a good volum e form on A =G which willthen localize to a volum e

form on M I. In the work ofDonaldson the cohom ology classes considered for

this purpose are certain characteristic classes (ofthe universalbundle of[39])

which also arise naturally in the �eld theoretic description [40,9].Forinstance,

oneofthebuilding blocksisthe two-form � asgiven by (4.23)which represents

thecurvatureform � A (4.14).Unfortunately,theseintersection num bersarevery

di�cultto calculatein general.Fordetailspleaseconsultthecited literature.

4.3 Flat connections in tw o and three dim ensions

Itis,ofcourse,also possibleto turn around thestrategy oftheprevioussection,

i.e.to start with the M athai-Quillen form alism applied to som e vector bundle

over A =G and to then reconstruct the action ofthe corresponding topological

gaugetheory from there.

28

Letus,forinstance,considertheproblem ofconstructing a topologicalgauge

theory in 3d whose partition function (form ally)calculatesthe Eulercharacter-

istic�(M 3)ofthem odulispaceM 3 = M 3(M ;G)of atG connectionson som e

three-m anifold M .W eactuallyalreadyknow twowaysofachievingthis,provided

thatwecan �nd avector�eld v on A 3=G3 (thesuperscriptsarearem inderofthe

dim ension we are in)whose zero locusisM 3.Fortuitously,in three dim ensions

such a vector�eld exists,nam ely v = �FA. A priori,thisonly de�nes a vector

�eld on A 3,as�FA 2 1(M ;g). Itis,however,horizontal(d�A � FA = 0 by the

Bianchiidentity dAFA = 0)and thusprojectsto a vector�eld on A 3=G3 whose

zerolocusisM 3.Thisvector�eld isthegradientvector�eld oftheChern-Sim ons

functional

CS(A)=

Z

M

AdA + 2

3A3; (4.26)

whose criticalpointsarewellknown to be the atconnections.(Ofcourse,this

doesnotreally de�ne a functionalon A 3=G3,asitchangesby a constanttim es

thewinding num berunderlargegaugetransform ations.Butitsderivativeiswell

de�ned and thisnon-invarianceim pliesthattheone-form dA CS(A)passesdown

to a closed butnotexactone-form F A on A 3=G3.Explicitly,F A isgiven by

F A :T[A ]A3=G3 ! R

[X ] !

Z

M

FA X : (4.27)

Notethatthisdoesnotdepend on therepresentativeof[X ]asR

M FAdA�= 0.) In

two dim ensionssuch a vector�eld appearsnotto existat�rstsightand onehas

to bea littlem oreinventive(cf.[17]and therem arksattheend ofthissection).

Given thisvector�eld,the�rstpossibility isthen toadapttheAtiyah-Je�rey

construction oftheprevioussection to thecaseX = A 3=G3 and E = T(A 3=G3),

to usev = �FA astheregularizing section for

�v(A3=G3)= �(M 3) ; (4.28)

and to representthisby thefunctionalintegral

�(M 3)=

Z

A 3=G3ev;r (A

3=G3) : (4.29)

Ofcourse,the ‘action’,i.e.the exponentof(4.29),willcontain non-localterm s

like the curvature tensor R A =G (4.7),as in (4.24). As this is undesirable for a

fundam entalaction,we willintroduce auxiliary �elds (like those we elim inated

in going from (4.17)to (4.24))to rewritetheaction in localform .

Alternatively,wecanconstructsupersym m etricquantum m echanicsonA 3=G3

using _A + v asthe section ofT(LA 3=G3),i.e.we use the action SM ;W (3.15)of

section 3 and substitute M ! A 3=G3 and W ! CS(A). This willgive us a

(non-covariant)(3+ 1)-dim ensionalgaugetheory on M � S1 (in fact,the(3+ 1)-

decom position ofDonaldson theory,see [41,1]and [18]for details). However,

29

from thegeneralargum entsofsection 3 weknow thatonly theconstantFourier

m odes willcontribute, so that one is left with an e�ective three-dim ensional

action which isidenticalto theoneobtained by the�rstm ethod.

Irrespectiveofhow onechoosestogoaboutconstructing theaction (thereare

stillfurtherpossibilities,seee.g.[13,14,17]),itreads

SM =

Z

M

�

B 1FA + �B 1 � B1=2+ dAu� dAu=2� dA�� � dA� + � dA

�

+�

u[ ;�� ]+ �dA � + ��dA � � + ��[ ;� ]� ��[� ;�� ]=2�

: (4.30)

u isascalar�eld,and asin supersym m etricquantum m echanicswehavedenoted

the �eld � ofthe M athai-Quillen form ula by � . The restshould look fam iliar.

Super�cially,this action is very sim ilar to that ofDonaldson theory. There is

a Gaussian constraintonto atconnections,the tangents have to satisfy the

linearized atnessequations,and therearecubicinteraction term sinvolving the

scalar�elds�;�� and u.However,thereisoneim portantdi�erence,nam ely that

there isa perfectsym m etry between and � . Asin supersym m etric quantum

m echanics,both representtangentvectors,wealso seethatboth aregauge�xed

to be horizontal,and both have to be tangentto M 3. In particular,therefore,

therewillbean equalnum berof and � zero m odesand wehavethepossibility

ofobtaining a non-zero result even ifdim (M 3) 6= 0. This is reassuring as we,

afterall,expectto �nd Z(SM )= �(M 3). Letusnow show thatthisisindeed

thecase.

� Firstofallintegration over� and �� forces and� tobehorizontal,hA =

,hA � = � ,i.e.to representtangentvectorsto A 3=G3

� Setting � = 1,integration over�� yields� = �G0A � [ ;� ],giving riseto a

term

h�[� ;�� ];G 0

A � [ ;� ]i=2

in theaction

� Theequation ofm otion foru reads

u = G0

A � [ ;�� ]

and plugging thisback into theaction oneobtainsa term

h�[ ;�� ];G 0

A � [ ;�� ]i=2

� Thiscom bination ofGreensfunction isprecisely thatappearing in thefor-

m ula (4.7)forthe Riem ann curvature tensorR A =G.Thus we have already

reduced the action to the form SM = R A =G + ‘som ething0 and we expect

the‘som ething’to bethecontribution (4.10)to R M (4.8)quadraticin the

extrinsiccurvatureK M .

30

� To evaluate the integraloverthe rem aining �eldsA, ,and � we expand

them abouttheirclassicalcon�gurationswhich wecan taketo be atcon-

nectionsA c and theirtangents(because of�-independence). By standard

argum ents we m ay restrict ourselves to a one-loop approxim ation and to

thisordertherem aining term sin theaction becom e

Z

M

(dA cA q � dA c

A q=2+ [� c; c]A q) :

� Finally,integration overAq yields

h[� c; c];G2

A c[� c; c]i=2 ;

which we recognize to be precisely the contribution (4.10) ofK M . Thus

we have reduced the action (4.30)to R M ,expressed in term softhe clas-

sicalcon�gurations A c, c and � c. W e are now on fam iliar ground (see

e.g.(2.25,3.20))and know that evaluation ofthis �nite dim ensionalinte-

gralgives

Z(SM )= �(M ) : (4.31)

Thiscalculationalsoillustrateshow theGauss-Codazziequationsem ergefrom the

M athai-Quillen form in general.Guided bythisexam pleitisnow straightforward

toperform theanalogousm anipulationsin the�nitedim ensionalcase(section 2)

and in supersym m etric quantum m echanics(section 3).

W eend this3d exam plewith therem ark that,by a resultofTaubes[26],the

partition function of(2.24)form ally equalstheCasson invariantofM ifM isa

hom ology three-sphere [13]. This,com bined with the above considerations,has

led usto propose�(M )asa candidateforthede�nition oftheCasson invariant

ofm oregeneralthree-m anifolds(see[17]forsom eprelim inary considerations).

The sim plestexam ple to considerin two dim ensionsisthe analogueofDon-

aldson theory,i.e. a topological�eld theory describing intersection theory on a

m odulispaceM 2 of atconnectionsin twodim ensions.Instead ofthebundleE+with standard �bre 2

+(M ;g)(4.16)wechoosethebundleE0 (4.15)with standard

�bre 0(M ;g).Thiswillhavethee�ectofreplacing theself-dualtwo-form sB +

and �+ ofDonaldson theory by zero-form sB 0 and �0. A naturalsection ofE0iss(A)= �FA with zero locusM 2.Thisresultsin the trading of(FA)+ and its

linearization (dA )+ forFA and itslinearization dA in theaction (4.17).W ith

this dictionary in m ind the action is precisely the sam e as that ofDonaldson

theory. Itisalso the 2d version of(4.22)and we have thus justcom pleted the

construction of

Exam ple 5 X = A 2=G2,E = E0,s= �FAThefundam entalreason forwhy thistheory isso sim ilarto Donaldson theory is

thatinbothcasesthedeform ationcom plexisshortsothatonewill�ndessentially

thesam e�eld content.In threedim ensions,on theotherhand,thedeform ation

31

com plex islongerby oneterm and thisisre ected in theappearanceofthescalar

�eld u in (4.30).

Again,the partition function,i.e.the regularized Euler num ber ofE0,will

vanish when dim (M 2) 6= 0. But,none too surprisingly,there also exist ana-

loguesoftheDonaldson polynom ials,theobservablesofDonaldson theory,which

com e to the rescue in this case. Life in two dim ensions is easier than in four,

and thecorresponding intersection num bershaveindeed been calculated recently

by Thaddeus [42]using powerfultools ofconform al�eld theory and algebraic

geom etry (seealso [43,44]).

Asour�nalexam ple letusconsidera topologicalgauge theory representing

theEulercharacteristicofM 2.Asm entioned above,�FA isnota vector�eld on

A 2=G2,so thatitisnotim m ediately obviouswhich section ofTA 2=G2 to choose.

Thedim ensionalreduction oftheaction (4.30)suggests,thattherightbasespace

to considerisX = A 2 � 0(M ;g),where the second factorrepresentsthe third

com ponent� ofA.Then apossiblesection ofTX isV (A;�)= (�dA�;�FA)whose

zerolocus(forirreducibleA)isindeed precisely thespaceof atconnections.But

thisisnotthe com plete story yet. The problem is,that�dA� isonly horizontal

ifA is at(d�A � dA� = [�FA;�]).Thus,onepossibility isto usea delta function

instead ofa Gaussian constraintonto atconnections(� = 0). Thisaction can

befound in [17].Alternatively,onem ightattem ptto replace �dA� by hA � dA�.

Thisnecessitatesthe introduction ofadditionalauxiliary �eldsto elim inate the

non-locality ofhA,and a m oredetailed investigation ofthispossibility isleftto

thereader.

A cknow ledgem ents

Iwish to thank R.Gielerak and thewholeorganizing com m itteeforinviting m e

to lectureatthisSchool.Thanksarealso dueto alltheparticipantsforcreating

such astim ulatingatm osphere,andtoJaapKalkm an forsendingm e[23].Finally,

Iwish to acknowledge the�nancialsupportoftheStichting FOM .

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