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-Simonsgaugetheorysinceitsrelationto

2、knottheorywasdiscoveredtenyearsagoispresented.EmphasisismadeontheanalysisoftheperturbativestudyofthetheoryanditsconnectiontothetheoryofVassilievinvariants.ItisdescribedhowthestudyofthequantumeldtheoryforthreedierentgaugexingsleadstothreedierentrepresentationsforVassilievinvariants.Twooftheseg

3、augexingsleadtowellknownrepresentations:thecovariantLandaugaugecorrespondsarXiv:hep-th/9905057v18May1999tothecongurationspaceintegralswhilethenon-covariantlight-conegaugetotheKontsevichintegral.Theprogressmadeintheanalysisofthethirdgaugexing,thenon-covarianttemporalgauge,isdescribedindetail.In

4、thiscaseoneobtainscombinatorialexpressions,insteadofintegralones,forVassilievinvariants.Theapproachbasedonthislastgaugexingseemsverypromisingtoobtainafullcombinatorialformula.WecollectthecombinatorialexpressionsforalltheVassilievinvariantsuptoorderfourwhichhavebeenobtainedinthisapproach.1Invited

5、lecturedeliveredattheworkshopTrendsinTheoreticalPhysicsII",heldatBuenosAires,Argentina,November29{December5,1998.1IntroductionTheconnectionbetweenChern-SimonsgaugetheoryandthetheoryofknotandlinkinvariantswasestablishedbyEdwardWittentenyearsago[1].Sincethenthetheoryhasbeenstudiedfromavarietyofpoi

6、ntsofview.Manyofthestandardmethodsineldtheoryhavebeenapplied,generatingresultswhichbecameimportantinthedevelopmentofknottheory.Theinterplaybetweenquantumeldtheoryandknottheoryhasbeenveryrichinbothdirections.Thoughtheresultshavebeenmorespectacularintheknottheorydirection,onemustnotforgetthatthed

7、evelopmentsinChern-Simonsgaugetheoryhaveconstitutedaconstanttestofourknowledgeinquantumeldtheory.Infact,ithasbeenfoundthatnotalwaysthequantumeldtheorymethodshavebeenabletoprovidetherightanswer.Asitwillbedescribedinde