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1、S´eminaireBourbakiJuin200052`emeann´ee,1999-2000,no875VERTEXALGEBRASANDALGEBRAICCURVESbyEDWARDFRENKEL1.INTRODUCTIONVertexoperatorsappearedintheearlydaysofstringtheoryaslocaloperatorsdescribingpropagationofstringstates.Mathematicalanaloguesoftheseoperato
2、rswerediscoveredinrepresentationtheoryofaffineKac-MoodyalgebrasintheworksofLepowsky–Wilson[LW]andI.Frenkel–Kac[FK].Inordertoformalizetheemergingstructureandmoti-vatedinparticularbytheI.Frenkel–Lepowsky–MeurmanconstructionoftheMoonshineModuleoftheMonstergr
3、oup,Borcherdsgavethedefinitionofvertexalgebrain[B1].Thefoundationsofthetheoryweresubsequentlylaiddownin[FLM,FHL];inparticular,itwasshownin[FLM]thattheMoonshineModuleindeedpossessedavertexalgebrastructure.Inthemeantime,Belavin,PolyakovandZamolodchikov[BPZ
4、]initiatedthestudyoftwo-dimensionalconformalfieldtheory(CFT).VertexalgebrascanbeseeninretrospectasthemathematicalequivalentofthecentralobjectsofCFTcalledthechi-ralsymmetryalgebras.Moreover,thekeypropertyofassociativityofvertexalgebrasisequivalenttothepro
5、pertyofoperatorproductexpansioninCFT,whichgoesbacktothepioneeringworksofPolyakovandWilson.Thus,vertexalgebrasmaybethoughtofasthemathematicallanguageoftwo-dimensionalconformalfieldtheory.Vertexalgebrashaveturnedouttobeextremelyusefulinmanyareasofmathemati
6、cs.Theyarebynowubiquitousinrepresentationtheoryofinfinite-dimensionalLiealgebras.Theyhavealsofoundapplicationsinsuchfieldsasalgebraicgeometry,theoryoffinitegroups,modularfunctionsandtopology.RecentlyBeilinsonandDrinfeldhaveintroducedaremarkablegeometricver
7、sionofvertexalgebraswhichtheycalledchiralalgebras[BD3].Chiralalgebrasgiverisetosomenovelconceptsandtechniqueswhicharelikelytohaveaprofoundimpactonalgebraicgeometry.Inthistalkwereviewthetheoryofvertexalgebraswithaparticularemphasisontheiralgebro-geometri
8、cinterpretationandapplications.Westartin§2withtheaxiomaticdefinitionofvertexalgebra,whichissomewhatdifferentfrom,butequivalenttoBorcherds’875-02originaldefinition(see[DL,FKRW,K2]).Wethendiscusssomeoftheirmostimportantpropertiesandgi
