Vertex algebras and the formal loop space

arXiv:math/0107143v3[math.AG]9Jul2004VERTEXALGEBRASANDTHEFORMALLOOPSPACEM.KapranovandE.Vasserot1.IntroductionOneofthesalientmathematicalfeaturesofstringtheoryistheimportanceofvertexalgebras.TheirroleinthetheorycanbecomparedtothatofLiealgebrasinthe“ordinary”physicsofpointparticles.Mathematically,theapproachofstringtheorycanbecastintermsofanalysisonthespaceoffreeloops,i.e.,smoothmapsS1→XwhereXisagiven“spacetime”manifold.Accordingly,onehasthefolkloreprinciplethatconstructionsinvolvingthespaceoffreeloopsleadtovertexalgebras.Oneclassofsuchconstructionsisprovidedbythespacesofhighestweightrepresentationsofloopgroups.AnotherisΩchX,thechiraldeRhamcomplexofanalgebraicvarietyX,introducedbyMalikov,SchechtmanandVaintrob[MSV].Heuristically,thiscomplexshouldbeinterpretedintermsofLX,thespaceoffreeloopsanditssubvarietyL0Xconsistingofloopsextendingholomorphicallyintotheunitdisk.Moreprecisely,ΩchXcanbethoughtofasthesemiinfinitedeRhamcomplexwithcoefficientsinthespaceofdistri-butionsonLXsupportedonL0X.Thisisnot,however,thewayΩchXhasbeendefinedmathematically.Thedefinitiongivenin[MSV]isofmorecomputationalnatureandproceedsbyconstructingtheactionofthegroupofdiffeomorphismsontheirreduciblemoduleovertheHeisenbergalgebra.Inthatapproachitseemsmiraculousthatsuchanactionexistsatall.Theaimofthispaperistwofold.First,togiveaprecisemathematicaltheoremunderlyingtheabovefolkloreprincipleaboutvertexalgebras.Forthis,weintro-duceanalgebro-geometricversionofthefreeloopspaceL(X)foranyschemeXoffinitetypeoverafield.Thisisanind-schemecontainingL0(X),theschemeofformalgermsofcurvesonXstudiedin[DL].WeprovethatbothL(X)andL0(X)themselvespossessanon-linearversionofthevertexalgebrastructure(whichmakesitclearthatanynaturallinearconstructionappliedtothemshouldgiveavertexTypesetbyAMS-TEX12M.KAPRANOVANDE.VASSEROTalgebraintheusualsense).Moreprecisely,weusethegeometricapproachtover-texalgebrasdevelopedbyBeilinsonandDrinfeld[BD1]andbasedontheconceptsofchiralalgebrasandfactorizationalgebras.Thelatterconcepthasanaturalnonlinearversion,thatofafactorizationmonoid.Whatweproveisthatnatural“global”versionsofL(X),L0(X)havenaturalstructuresoffactorizationmonoids.AnearlierknownexampleofafactorizationmonoidisgivenbytheaffineGrass-mannian[G],andthisexplainswhythespacesofrepresentationsofloopgroupsarevertexalgebras.Ourconstructionissimilarinspirit.TogiveagooddefinitionofthealgebraicanalogofthefullloopspaceLXonehastoovercomeacertainsubtlety.Namely,anaturalapproachwouldbetotryto(ind-)representafunctorwhichtoanycommutativeringRassociatesthesetofR((t))-pointsofX.(ThisisexactlyhowonedefinestheschemeL0(X),withR[[t]]insteadofR((t)).)IfXisaffine,thisindeedgivesagoodind-schemewhichwedenote˜L(X).ButwhenXis,say,projective,then(forRafield)thereisnodifferencebetweenR[[t]]-pointsandR((t))-pointsofX(valuativecriterionofproperness),soitmayseemthatnothingisgainedbyallowingLaurentseries.Tostatethisphenomenondifferently,theind-schemes˜L(U)foraffineU⊂Xdonotgluetogetherwell.Thisisinfactunderstandableongeneralgrounds:theloopspaceLXisnottheunionoftheLUsincealoopneednotspendallitstimeinanygivenU.Togetaroundthisdifficultyweadoptthefollowingstrategy.ForanaffineXweconsiderL(X),theformalneighborhoodofL0(X)in˜L(X).Sowearedealingwithformalloopswhichare“infinitesimalintheLaurentdirection”.Then,weprovethattheL(U),U⊂X,doindeedpossesstherightgluingproperties.Thisisduetotheinfinitesimalnatureofourloops.TheroleofnilpotentthickeningsinLaurentseriesmodelsforloopspaceswasfirstpointedoutbyC.Contou-Carr`ere[CC]whowasstudying,inournotation,thegroupind-scheme˜L(Gm)andfoundthatitisanontrivialformalthickeningofL0(Gm)×Z.OursecondgoalistogiveadirectgeometricconstructionofΩchX(forsmoothX)intermsofourmodelfortheloopspace.Bytheabove,thisconstructionexplainsalsothefactthatΩchXisasheafofvertexalgebras.Inordertoachievethis,werepresentL(X)asanind-pro-objectinthecategoryofschemesoffinitetypeandthenshowthattheshifteddeRhamcomplexesofthetermsofthisind-pro-systemarrangenaturallyintoadoubleinductivesystemwhoseinductivelimitisidentifiedwithΩchX.Aswiththestudyofformalarcsandmotivicintegration[DL],onecanviewourconsiderationsasalgebro-geometricanalogsofthebasicconstructionsofp-adicanalysis.Thedifferencebetweenourind-schemeL(X)andthemorefamiliarschemeVERTEXALGEBRASANDFORMALLOOPS3L0(X)issimilartothedifferencebetweenQpandZp:whilethelatterisapro-objectinthecategoryofschemesoffinitetype(resp.finitesets),theformerisanind-pro-object.Further,ourapproachtoΩchXissimilartotheconstructionofthespaceoflocallyconstantfunctionswithcompactsupportonQp=lim−→ilim←−jp−iZp/pjZpasthedoubleinductivelimitofthespacesoffunctionsonthefinitesetsp−iZp/pjZp,cf[P].Noticethatthereasonthatthesespacesoffunctionsindeedformadoubleinductivesystem(withrespecttothemapsofinverseimageinthej-directionanddirectimageinthei-direction)isthatthecommutativesquaresintheind-pro-systemp−iZp/pjZpareCartesian(sothatwehavebasechange).ThisisanalgebraiccounterpartofthepropertyofthelocalcompactnessofQp,see[Kat].Inoursituationitisequallyimportantthattheind-schemeL(X)satisfiesacertainformalanalogoflocalcompactness.Thisworkhasbeendo