An Introduction to Chern-Simons Theory

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1、AnIntroductiontoChern-SimonsTheory1BasicDenitionsRecallthataprincipalG-bundleEoverBisabrationwithbrebeingaLiegroupGandthereisafreeright(resp.left)actiononthetotalspacebyelementsofGsuchthatthisisfreeandtransitiveoneachbre.Themap:E!Binducesd:TE!TB,whereTeElocallysplitsinto

2、T(e)BT1Gbylocaltriviality.Thus,kerdisT1Gwhichcanbecanonicallyidentiedwithg.Forsuchabundle,aconnectionisa1-formonEtakingvaluesingandsatisfying(1.1)R=(ad)1gg(1.2)(v)=vg;v2kerd:HeretherightactionRgactsbygontherightandvgreferstotheidenticationofvwithanelementvgofgasme

3、ntionedbefore.Inparticular,thismeansthatifxdenotetheMaurer-CartanformonEx=Gandix:Ex,!E,thenix()=x.Alsorecallthestructureequation1(1.3)dx+[x;x]=0:2ItfollowsfromelementaryobstructiontheorythatProposition1.1.LetE!BbeasabovesuchthatGisconnected,simplyconnectedandcompact.Let

4、Bbeamanifoldofdimensionatmost3.Thenthebundleistrivial.Inwhatfollows,letMbeaclosed,connectedandoriented3-manifoldandletEbeaprincipalSU(2)-bundleonM.WexatrivializationonE.Leti(M;su)=(M;^iTM(Msu))22denotethespaceofformsonMwithvaluesinsu2.Therearetwoobviouswaystoextenda21(M;su

5、2)toa1-formonE=MSU(2):(1)using(1.1)anddeclaringittobetheMaurer-Cartanformoneachbre,(2)justextendtoeachcopyofMfggby(1.1).Thus,thespaceofconnectionsonany(principal)SU(2)-bundleoverM(makinguseof(1)above)isjustA(M)=1(M;su2)-thespaceofsu2-valued1-formsonM.Thecurvatureofaconnecti

6、ona2A(M)isdenedtobe(1.4)F=da+a^a22(M;su):a2Weobserveherethat1
1(a^a)(X;Y)=a(X)a(Y)a(Y)a(X)=[a(X);a(Y)]221and(1.5)RF=adgag1Fa(1.6)iF=0:xaThelastequationfollowsfrom(1.3).Wecallaconnection atifthecurvatureiszero.TheBianchiidentityfollowsbydierentiatingFa:(1.7)dFa+[a;Fa]=0:

7、Introducethecovariantderivative(1.8)da=d+ad(a):Thisinducesthecomplex0da1da2da3(1.9)(M;su2)!(M;su2)!(M;su2)!(M;su2):Thefailureoftheabovetobeachaincomplex,i.e.,da2=0,ismeasuredbythe atnessofasinceda2=Fa.22TheChern-SimonsfunctionalDenition2.1.Fora2A(M),chooseanoriented4-manifold

8、WwhichboundsMandcollarneighbourhoodoftheboundary@WU=M[0;1]: