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1、DAMTP-1999-143REVIEWARTICLEAnIntroductiontoConformalFieldTheoryMatthiasRGaberdielzDepartmentofAppliedMathematicsandTheoreticalPhysics,SilverStreet,Cambridge,CB39EW,UKandFitzwilliamCollege,Cambridge,CB30DG,UKAbstract.Acomprehensiveintroductiontotwo-dimensionalconformaleldtheoryisgiven.PAC
2、Snumbers:11.25.HfSubmittedto:Rep.Prog.Phys.arXiv:hep-th/9910156v21Nov1999zEmail:M.R.Gaberdiel@damtp.cam.ac.ukConformalFieldTheory21.IntroductionConformaleldtheorieshavebeenatthecentreofmuchattentionduringthelastfteenyearssincetheyarerelevantforatleastthreedierentareasofmoderntheoretica
3、lphysics:conformaleldtheoriesprovidetoymodelsforgenuinelyinteractingquantumeldtheories,theydescribetwo-dimensionalcriticalphenomena,andtheyplayacentralr^oleinstringtheory,atpresentthemostpromisingcandidateforaunifyingtheoryofallforces.Conformaleldtheorieshavealsohadamajorimpactonvariou
4、saspectsofmodernmathematics,inparticularthetheoryofvertexoperatoralgebrasandBorcherdsalgebras,nitegroups,numbertheoryandlow-dimensionaltopology.Fromanabstractpointofview,conformaleldtheoriesareEuclideanquantumeldtheoriesthatarecharacterisedbythepropertythattheirsymmetrygroupcontains,in
5、additiontotheEuclideansymmetries,localconformaltransformations,i.e.transformationsthatpreserveanglesbutnotlengths.Thelocalconformalsymmetryisofspecialimportanceintwodimensionssincethecorrespondingsymmetryalgebraisinnite-dimensionalinthiscase.Asaconsequence,two-dimensionalconformaleldthe
6、orieshaveaninnitenumberofconservedquantities,andarecompletelysolvablebysymmetryconsiderationsalone.Asabonadequantumeldtheory,therequirementofconformalinvarianceisveryrestrictive.Inparticular,sincethetheoryisscaleinvariant,allparticle-likeexcitationsofthetheoryarenecessarilymassless.Thi
7、smightbeseenasastrongargumentagainstanypossiblephysicalrelevanceofsuchtheories.However,allparticlesofany(two-dimensional)quantumeldtheoryareapproximatelymasslessinthelimitofhighenergy,andmanystructuralfeaturesofquantumeldtheoriesarebelievedtobeunchangedinthisappro