TheM athai-Quillen Form alism and TopologicalField Theory
Transcript of TheM athai-Quillen Form alism and TopologicalField Theory
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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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