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1、DAMTP-1999-143REVIEWARTICLEAnIntroductiontoConformalFieldTheoryMatthiasRGaberdielzDepartmentofAppliedMathematicsandTheoreticalPhysics,SilverStreet,Cambridge,CB39EW,UKandFitzwilliamCollege,Cambridge,CB30DG,UKAbstract.Acomprehensiveintroductiontotwo-dimensionalconformal eldtheoryisgiven.PAC
2、Snumbers:11.25.HfSubmittedto:Rep.Prog.Phys.arXiv:hep-th/9910156v21Nov1999zEmail:M.R.Gaberdiel@damtp.cam.ac.ukConformalFieldTheory21.IntroductionConformal eldtheorieshavebeenatthecentreofmuchattentionduringthelast fteenyearssincetheyarerelevantforatleastthreedi erentareasofmoderntheoretica
3、lphysics:conformal eldtheoriesprovidetoymodelsforgenuinelyinteractingquantum eldtheories,theydescribetwo-dimensionalcriticalphenomena,andtheyplayacentralr^oleinstringtheory,atpresentthemostpromisingcandidateforaunifyingtheoryofallforces.Conformal eldtheorieshavealsohadamajorimpactonvariou
4、saspectsofmodernmathematics,inparticularthetheoryofvertexoperatoralgebrasandBorcherdsalgebras, nitegroups,numbertheoryandlow-dimensionaltopology.Fromanabstractpointofview,conformal eldtheoriesareEuclideanquantum eldtheoriesthatarecharacterisedbythepropertythattheirsymmetrygroupcontains,in
5、additiontotheEuclideansymmetries,localconformaltransformations,i.e.transformationsthatpreserveanglesbutnotlengths.Thelocalconformalsymmetryisofspecialimportanceintwodimensionssincethecorrespondingsymmetryalgebraisin nite-dimensionalinthiscase.Asaconsequence,two-dimensionalconformal eldthe
6、orieshaveanin nitenumberofconservedquantities,andarecompletelysolvablebysymmetryconsiderationsalone.Asabona dequantum eldtheory,therequirementofconformalinvarianceisveryrestrictive.Inparticular,sincethetheoryisscaleinvariant,allparticle-likeexcitationsofthetheoryarenecessarilymassless.Thi
7、smightbeseenasastrongargumentagainstanypossiblephysicalrelevanceofsuchtheories.However,allparticlesofany(two-dimensional)quantum eldtheoryareapproximatelymasslessinthelimitofhighenergy,andmanystructuralfeaturesofquantum eldtheoriesarebelievedtobeunchangedinthisappro
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