introduction to vertex algebras

《introduction to vertex algebras》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、DESY93-120hep-th/9308151IntroductiontoVertexAlgebras,BorcherdsAlgebras,andtheMonsterLieAlgebraReinholdW.GebertyIIndInstituteforTheoreticalPhysics,UniversityofHamburgLuruperChaussee149,D-22761Hamburg,GermanyAugust25,1993Thetheoryofvertexalgebrasconstitutesamathem

2、aticallyrig-orousaxiomaticformulationofthealgebraicoriginsofconformaleldtheoryInthiscontextBorcherdsalgebrasariseascertainphysical"subspacesofvertexalgebras.Theaimofthisreviewistogiveapedagogicalintroductionintothisrapidly-developingareaofmathematics.Basedonthe

3、machineryofformalcalculuswepresenttheaxiomaticdenitionofvertexalgebras.WediscusstheconnectionwithconformaleldtheorybyderivingimportantarXiv:hep-th/9308151v23SEP93implicationsoftheseaxioms.Inparticular,manyexplicitcalcula-tionsarepresentedtostresstheeminentroleo

4、ftheJacobiidentityaxiomforvertexalgebras.Asaclassofconcreteexamplesthever-texalgebrasassociatedwithevenlatticesareconstructedanditisshownindetailhowaneLiealgebrasandthefakeMonsterLiealgebranaturallyappear.Thisleadsustotheabstractdef-initionofBorcherdsalgebrasasg

5、eneralizedKac-Moodyalgebrasandtheirbasicproperties.Finally,theresultsaboutthesimplestgenericBorcherdsalgebrasareanalysedfromthepointofviewofsymmetryinquantumtheoryandtheconstructionoftheMonsterLiealgebraissketched.toappearinInt.J.Mod.Phys.ySupportedbyKonrad-Aden

6、auer-Stiftunge.V.1IntroductionNowadaysmosttheoreticalphysicistsareawareofthefactthatthepresentrelationbetweenmathematicsandphysicsischaracterizedbyanincreasingoverlapbetweenthem.ProminentexamplesforthatdevelopmentarethevertexoperatorconstructionofKac-Moodyalgebra

7、s,Calabi-Yaumanifoldsaspossiblegroundstatesofstringtheory,thesumoverRiemannsur-facesformulationofstringtheory,theimportantroleofmodularformsandthetafunctionsinconformaleldtheory,theMoonshine"meromorphicconformaleldtheoryasanaturalrep-resentationspacefortheMons

8、tersporadicsimplegroup.Thesefewexamplesshowsucientlythatitisneccessarytolearnfromeachotherandtotransfertoolsandtechniques.Inthisworkweareconcernedwiththemathe