labastida(1998)-chern-simons gauge theory-- ten years after

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1、USC-FT-7/99hep-th/9905057CHERN-SIMONSGAUGETHEORY:TENYEARSAFTER1J.M.F.LabastidaDepartamentodeFsicadePartculasUniversidadedeSantiagodeCompostelaE-15706SantiagodeCompostela,Spaine-mail:labasti@fpaxp1.usc.esAbstractAbriefreviewontheprogressmadeinthestudyofChern-Si

2、monsgaugetheorysinceitsrelationtoknottheorywasdiscoveredtenyearsagoispresented.EmphasisismadeontheanalysisoftheperturbativestudyofthetheoryanditsconnectiontothetheoryofVassilievinvariants.Itisdescribedhowthestudyofthequantumeldtheoryforthreedierentgaugexingslea

3、dstothreedierentrepresentationsforVassilievinvariants.Twoofthesegaugexingsleadtowellknownrepresentations:thecovariantLandaugaugecorrespondsarXiv:hep-th/9905057v18May1999tothecongurationspaceintegralswhilethenon-covariantlight-conegaugetotheKontsevichintegral.Th

4、eprogressmadeintheanalysisofthethirdgaugexing,thenon-covarianttemporalgauge,isdescribedindetail.Inthiscaseoneobtainscombinatorialexpressions,insteadofintegralones,forVassilievinvariants.Theapproachbasedonthislastgaugexingseemsverypromisingtoobtainafullcombinator

5、ialformula.WecollectthecombinatorialexpressionsforalltheVassilievinvariantsuptoorderfourwhichhavebeenobtainedinthisapproach.1InvitedlecturedeliveredattheworkshopTrendsinTheoreticalPhysicsII",heldatBuenosAires,Argentina,November29{December5,1998.1IntroductionTheco

6、nnectionbetweenChern-SimonsgaugetheoryandthetheoryofknotandlinkinvariantswasestablishedbyEdwardWittentenyearsago[1].Sincethenthetheoryhasbeenstudiedfromavarietyofpointsofview.Manyofthestandardmethodsineldtheoryhavebeenapplied,generatingresultswhichbecameimportant

7、inthedevelopmentofknottheory.Theinterplaybetweenquantumeldtheoryandknottheoryhasbeenveryrichinbothdirections.Thoughtheresultshavebeenmorespectacularintheknottheorydirection,onemustnotforgetthatthedevelopmentsinChern-Simonsgaugetheoryhaveconstitutedaconstanttestof

8、ourknowledgeinquantumeldtheory.Infact,ithasbeenfoundthatnotalwaysthequantumeldtheorymethodshavebeenabletoprovidetherightanswer.Asitwillbedescribedinde