On the D-module and formal variable approaches to

arXiv:q-alg/9603020v123Mar1996OntheD-moduleandformal-variableapproachestovertexalgebrasYi-ZhiHuangandJamesLepowsky1IntroductionInaprogramtoformulateanddeveloptwo-dimensionalconformalfieldthe-oryintheframeworkofalgebraicgeometry,BeilinsonandDrinfeld[BD]haverecentlygivenanotionof“chiralalgebra”intermsofD-modulesonalgebraiccurves.Thisdefinitionconsistsofa“skew-symmetry”relationanda“Jacobiidentity”relationinacategoricalsetting,anditleadstotheoperatorprod-uctexpansionforholomorphicquantumfieldsinthespiritoftwo-dimensionalconformalfieldtheory,asexpressedin[BPZ].Becausethisoperatorprod-uctexpansion,properlyformulated,isknowntobeessentiallyavariantofthemainaxiom,the“Jacobiidentity”[FLM],forvertex(operator)algebras([Borc],[FLM];see[FLM]fortheproof),thechiralalgebrasof[BD]amountessentiallytovertexalgebras.Inthispaper,weshowdirectlythatthechiralalgebrasof[BD]arees-sentiallythesameasvertexalgebraswithoutvacuumvector(andwithoutgrading),byestablishinganequivalencebetweentheskew-symmetryandJa-cobiidentityrelationsof[BD]andthe(similarly-named,butdifferent)skew-symmetryandJacobiidentityrelationsintheformal-variableapproachtovertexoperatoralgebratheory(see[FLM],[FHL]).Inparticular,amongtheequivalentformulationsofthenotionofvertex(operator)algebra,theD-modulenotionofchiralalgebracorrespondsthemostcloselytotheformal-variablenotion,ratherthanto,say,theoperator-product-expansionnotion(basedonthe“commutativity”and“associativity”relations,asexplainedin[FLM],[FHL])ortothegeometricoroperadicnotion([Hu1],[Hu2],[HL1]).Moreprecisely,weprovethatforanynonemptyopensubsetXofC,thecategoryofvertexalgebraswithoutvacuumoverX(seeDefinitions2.21and5.1)andthecategoryofchiralalgebrasoverX(seeDefinition4.1)areequivalent(seeSection5).Beilinson-Drinfeld’snotionofchiralalgebraisingeneralformulatedoverhigher-genuscurves.Butsincechiralalgebrasinthissenseareessentiallylocalobjects,theequivalenceprovedinthispapershowsthatthenotionin[BD]isindeedessentiallyequivalenttothenotionofvertexalgebrawithoutvacuum.Wehopethatthepresentexpositoryexercisehelpstoilluminatethere-lationsbetweenthetheoriesandphilosophiesofD-modulesandofvertexoperatoralgebras.Forexample,intheJacobiidentityforvertexalgebras,thethreeformalvariablesareonequalfootingbecauseofanintrinsicS3-symmetry(see[FHL],Section2.7),whileintheD-moduleapproach,thereareonlytwo(complex)variables,asintheoperator-product-expansionap-proach.(SeeRemark3.17.)ToseetheS3-symmetryexplicitlyinthealgebro-geometricframework,wewouldhavetointroduceananalogueofthenotionofD-moduleallowingglobaltranslationsofavariableratherthanjust“in-finitesimaltranslations.”TheJacobiidentitywouldthenbeinterpretedasanidentityintermsofsuch“modifiedD-modules,”sothatthethreevariablesinvolvedwouldplaysymmetricroles.Thiswillbediscussedinfuturepublica-tions.TheJacobiidentityforvertexoperatoralgebrasanditsS3-symmetryinfactplayacentralroleinthetheoryofvertexoperatoralgebras,inpar-ticular,intheconstructionof“vertextensorcategories”(see[HL2],[HL3],[HL4]).Evenwithouttheintroductionofsuchglobaltranslationsofvariables,allofthemanycalculationsinthispaperinvolvingbinomialexpansionscanbegreatlysimplifiedifwesystematicallyintroduceformal(notcomplex)variablesplayingtheroleof“formalglobaltranslations.”Forinstance,theexpression(z1−z2)n(A1⊗A2),n∈Z,occurringstartinginSection4canbeviewedasthecoefficientofx−n−1inx−1δ(z1−z2x)(A1⊗A2),wherexisaformalvariableandδ(z1−z2x)isdefinedinSection2.Inordertomakethisworkreasonablyself-contained,weincludeelemen-tarydefinitionsandnotionsneededinboththeories.Thereadercanconsult[FLM]and[FHL],forexample,forthemotivationanddevelopmentofthetheoryofvertexoperatoralgebras,and[Ha]forsheavesand[Borel]foralge-braicD-modules,whosetheorywasdevelopedbyBeilinsonandBernstein.Thispaperisorganizedasfollows:InSection2,werecallsomebasicnotationsandelementarytoolsandgivethedefinitionsofvertexalgebraandvertexalgebrawithoutvacuum.InSection3,werecallsomebasicconcepts2inthetheoryofD-modulesandgiveexampleswhichweshallneedlater.Beilinson-Drinfeld’snotionofchiralalgebraoverXforanonemptyopensubsetX⊂CisgiveninSection4.InSection5,wedefinethenotionofvertexalgebrawithoutvacuumoverXandprovetheequivalencetheoremstatedabove.WewouldliketothankP.DeligneandespeciallyA.Beilinsonforexplain-ingtheunpublishedwork[BD]tous.WearealsogratefultoF.Knop,whoseRutgerslecturenotesonD-modulesandrepresentationtheorywereveryhelpfultous.Y.-Z.H.issupportedinpartbyNSFgrantDMS-9596101andbyDIMACS,anNSFScienceandTechnologyCenterfundedundercontractSTC-88-09648,andJ.LbyNSFgrantDMS-9401851.2Vertexalgebrasandvertexalgebraswith-outvacuumFollowingthetreatmentin[FLM]and[FHL],wedescribethebasicnotationsandelementarytoolsneededtoformulatethenotionofvertexalgebra.WeworkoverC.Inthispaper,thesymbolsx,x0,x1,...areindependentcom-mutingformalvariables,andallexpressionsinvolvingthesevariablesaretobeunderstoodasformalLaurentseries.(Laterweshallalsousethesymbolsz,z1,...,whichwilldenotecomplexnumbers,notformalvariables.)Weusethe“formalδ-function”δ(x)=Xn∈Zxn,whichhasthefollowingsimpleandfundamentalproperty:ForanyLaurentpolynomi