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arXiv:hep-th/9210098v119Oct1992TPJU-7/92NoncommutativeGeometryandGaugeTheoryonDiscreteGroups(revisedversion)AndrzejSitarz1DepartmentofFieldTheoryInstituteofPhysics,JagiellonianUniversityReymonta4,PL-30-059Krak´owPolandAbstract:Webuildandinvestigateapuregaugetheoryonarbitrarydiscretegroups.Asystematicapproachtotheconstructionofthedifferentialcalculusispresented.Wediscussthemetricpropertiesofthemodelsandintroducetheactionfunctionalsforunitarygaugetheories.AdetailedanalysisoftwosimplemodelsbasedonZ2andZ3follows.Finallywestudythemethodofcombiningthediscreteandcontinuousgeometry.1e-mail:ufsitarz@plkrcy11.bitnet1INTRODUCTIONANDNOTATIONThenoncommutativegeometryprovidesuswithafarmoregeneralframe-workforphysicaltheoriesthantheusualapproaches.Itsbasicideaistosubstituteanabstract,associativeandnotnecessarilycommutativealgebraforthealgebraoffunctionsonasmoothmanifold1−5.Thisallowsustousenontrivialalgebrasasageometricalsetupforthefield-theoreticalpurposes,inparticularforthegaugetheories,whichareofspecialinterestbothfrommathematicalandphysicalpointsofview.Theconstructionofnoncommutativegaugetheorieshasledtoaremark-ableresult,whichisthedescriptionoftheHiggsfieldintermsofagaugepotential.Thissuggestssomepossiblenontrivialgeometrybehindthestruc-tureoftheStandardModel.Severalexamplesofthiskind,withvariouschoicesoffundamentalobjectsofthetheory,havebeeninvestigatedinsuchcontext6−13.The”discretegeometry”models,whichtakeasthealgebrathesetoffunctionsonadiscretespace,seemtobeoneofthemostpromisinginterpretations4,5andsuggestthatsuchageometrymayplayanimportantroleinphysics.Recently,somemoreanalysishasbeencarriedoutfortwo-andthree-pointspaces15,16inthecontextofgrandunificationandgeneralrelativity.Weproposetodevelophereasystematicapproachtowardstheconstruc-tionofapuregaugetheoryonarbitrarydiscretegroups.ThechoiceofagroupasourbasespaceiscrucialforouranalysisandallowsustomakeuseofthecorrespondencewiththegaugetheoryonLiegroups.Theformalismoffinitederivationsandinvariantforms,whichweuseinourapproach,isequivalenttotheoneusedinvariousworks2,4,12,13forthetwo-pointspace,italsoextendsconsiderablyourearlierstudies14.Thepaperisorganizedasfollows:inthefirstsectionweconstructthetoolsofthedifferentialcalculus,thenweoutlinethegeneralformalismofgaugetheoriesinthiscaseandsomeproblemsoftheconstructionofactions.Thediscussionoftwoexamplesfollows.Finallywediscussotherpossibili-tiesoriginatingfromthesymmetryprinciplesandwepresentthemethodofcombiningthediscreteandcontinuousgeometry.12DIFFERENTIALCALCULUSLetGbeafinitegroupandAbethealgebraofcomplexvaluedfunctionsonG.Wewilldenotethegroupmultiplicationby⊙andthesizeofthegroupbyNG.TherightandleftmultiplicationsonGinducenaturalautomorphismsofA,RgandLg,respectively,(Rhf)(g)=f(g⊙h),(1)withasimilardefinitionforLg.NowwewillconstructtheextensionofAintoagradeddifferentialalge-bra.Weshallfollowthestandardprocedureofintroducingthedifferentialcalculusonmanifolds,inparticularonLiegroups.Thereforeweusealmostthesameterminology,althoughthedefinitionsofcertainobjectsmaydiffer.FirstletusidentifythevectorfieldsoverAwithlinearoperatorsonA,whichhavetheirkernelequaltothespaceofconstantfunctions.TheyformasubalgebraofGL(NG,C),withanadditionalstructureofafinitedimensionalleftmoduleoverA.Now,wecandefinethevectorspaceFofleftinvariantvectorfieldsassatisfyingthefollowingidentity:∂∈F⇔∀f∈ALh∂(f)=∂(Lhf).(2)BeforewediscussthealgebraicstructureofFletusobservethatthisvectorspaceisNG−1dimensionalanditgeneratesthemoduleofvectorfields.ThismeansthatforagivenbasisofF,∂i,i=1...NG,everyvectorfieldcanbeexpressedasalinearcombinationfi∂i,withthecoefficientsfifromthealgebraA.Fformsanalgebraitselfandwefindtherelationsofgeneratorstobeofthesecondorder,∂i∂j=XkCkij∂k,(3)whereCkijarethestructureconstants.Becauseoftheassociativityofthealgebratheymustobeythefollowingsetofrelations,XlClijCmlk=XlCmilCljk.(4)NowwechooseaspecificbasisofFandcalculatetherelations(3)inthisparticularcase.ItisconvenientforourpurposestointroducethebasisofF2labeledbytheelementsofG′=G\{e},whereeistheneutralelementofG.Furtheron,ifnotstatedotherwise,itshouldbeassumedthatallindicestakevaluesinG′.∂gf=f−Rgf,g∈G′,f∈A,(5)Thestructurerelations(3)becomeinthechosenbasisquitesimple,∂g∂h=∂g+∂h−∂(h⊙g),g,h∈G′.(6)AsanextstepletusintroducetheHaarintegral,whichisacomplexvaluedlinearfunctionalonAthatremainsinvariantundertheactionofRg,Zf=1NGXg∈Gf(g),(7)wherewenormalizedit,sothatR1=1.AlthoughtheelementsofFdonotsatisfytheLeibnizrule,theyareinversetotheintegration.Indeed,wenoticethatforeveryf∈Aandeveryv∈Ftheintegral(7)ofv(f)vanishes.ForthisreasonwecanconsiderthemascorrespondingtothederivationsonthealgebraA.Wedefinenowthespaceofone-formsΩ1asaleftmoduleoverA,whichisdualtothespaceofvectorfields.ItcouldbealsoconsideredasarightmodulewithanappropriatedefinitionoftherightactionofA.This,however,isastraightforwardconsequenceofthedifferentialstructureandwewilldiscussitlater.Nowwecanintroducethenotionofleftinvariantforms,which,whenactingontheelementsofF,giveconstantfunctions.HavingchosenthebasisofFweautomaticallyhavethedualbasisofF∗consistingofformsχg,
本文标题:Noncommutative Geometry and Gauge Theories on Disc
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