Combinatorics of free vertex algebras

arXiv:math/0103173v2[math.QA]8Jun2001CombinatoricsoffreevertexalgebrasMichaelRoitmanM.S.R.I.1000CentennialDr.Berkeley,CA94720E-mail:[e-mail]:Freevertexalgebras,latticevertexalgebras,conformalalgebrasINTRODUCTIONThispaperillustratesthecombinatorialapproachtovertexalgebra—studyofvertexalgebraspresentedbygeneratorsandrelations.Anecessaryingredientofthismethodisthenotionoffreevertexalgebra.Borcherds[2]wasthefirsttonotethatfreevertexalgebrasdonotexistingeneral.Thereasonforthisisthatvertexalgebrasdonotformavarietyofalgebras,asdefinedine.g.[4],becausethelocalityaxiom(see§2below)isnotanidentity.However,acertainsubcategoryofvertexalgebras,obtainedbyrestrictingtheorderoflocalityofgenerators,hasauniversalobject,whichwecallthefreevertexalgebracorrespondingtothegivenlocalitybound.In[15]somefreevertexalgebraswereconstructedandincertainspecialcasestheirlinearbaseswerefound.Inthispaperwegeneralizetheconstructionof[15]andfindlinearbasesofanarbitraryfreevertexalgebra.Itturnsoutthatfreevertexalgebrasarecloselyrelatedtothevertexalgebrascorrespondingtointegerlattices.Thelatteralgebrasplayaveryimportantroleindifferentareasofmathematicsandphysics.Theywereextensivelystudiedine.g.[5,6,9,11,14].Hereweexploretherelationbetweenfreevertexalgebrasandlatticevertexalgebrasinmuchdetail.Theseresultscomplywiththeuseoftheword“free”inphysicalliteraturereferingtosomeelementsoflatticevertexalgebras,likein“freefield”,“freebozon”or“freefermion”.Amongotherthings,wefindanicepresentationoflatticevertexalgebrasintermsofgeneratorsandrelations,thusgivinganalternativeconstructionofthesealgebraswithoutusingvertexoperators.Weremarkthatourconstructionworksinaverygeneralsetting;wedonotassumethelatticetobepositivedefinite,neithernon-degenerate,norofafiniterank.12MICHAELROITMANOrganizationofthemanuscriptWestartwithreviewingsomebasicdefinitionsofvertexandconformalalgebrain§1–§3.ThereadermayconsultKac’sbook[11]formoredetails.Thenin§4werecalltheconstructionofthevertexalgebraVΛcorrespondingtoanintegerlatticeΛ,thedetailscanbeagainfoundin[11].In§5weconstructthefreevertexalgebraFN(B)generatedbyasetBsothattheorderoflocalityN(a,b)ofapairofgeneratorsa,b∈Bisgivenbyanarbitrarysymmetricinteger-valuedfunctionN:B×B→Z.Theorem1describesasetT⊂FN(B)andclaimsthatTisalinearbasisofFN(B).Asacorollary,wefindthedimensionsofhomogeneouscomponentsofFN(B).In§11wewillprovethatTspansFN(B),andin§12wefinishtheproofofTheorem1byshowingthatTislinearlyindependent.In§6weconstructavertexalgebrahomomorphismϕ:FN(B)→VΛfromthefreevertexalgebraFN(B)tothevertexalgebracorrespondingtothelatticeΛ=Z[B].TheintegerformonΛisdefinedby(a|b)=−N(a,b)fora,b∈B.Theorem2thenstatesthatϕisinjective.Wewillprovethistheoremin§12.In§7,usingTheorem2,weproveaquantitativeversionofDong’slemma.In§8weapplythislemmatosettleaquestionraisedin[16]:weprovethatthelocalityfunctionofafreeconformalalgebrahasquadraticgrowth.In§9westudyhomogeneousconformalderivationsoffreevertexalge-bras.ItturnsoutthataparticularlyinterestingcaseiswhenthealgebraisgeneratedbyasingleelementasuchthatN(a,a)=−1.InthiscaseFN{a}
isembeddedintothefermionicvertexalgebraVZ.WeprovethatahomogeneouscomponentofϕFN({a})
⊂VZisanirreduciblelow-estweightmoduleovercertainconformalalgebracW⊂V0,suchthatthecoefficientalgebraofcWisacentralextensionoftheLiealgebraofdiffer-entialoperatorsonthecircle,see[7,11,17].Finally,in§10wefindapresentationoflatticevertexalgebrasintermsofgeneratorsandrelations,seeTheorem4.Itturnsoutthattherequiredrelationsareratherminimal.Ourproofiscompletelycombinatorial.ThefirststepistodeterminethestructureofthevertexalgebrainquestionasamoduleovertheHeisenbergalgebra.NotationsAllalgebrasandlinearspacesareoverafieldkofcharacteristic0.Herearesomeshortcutsusedthroughoutthepaper:{n∈Z|n0}=Z+,1k!n(n−1)···(n−k+1)=nk
,1k!Dk=D(k),Z(L)isthecenterofaLiealgebraL.COMBINATORICSOFFREEVERTEXALGEBRAS3AcknowledgementsThecoreofthisworkwasdonewhileIwasvisitingtheFieldsInstitute.IamverygratefultoorganizersoftheProgramonLieTheoryforinvitingmethere.IespeciallythankStephanBerman,YulyBilligandChongyingDongforhelpfuldiscussions.1.FIELDSANDLOCALITYLetV=V¯0⊕V¯1beavectorsuperspaceoverk.TakeaformalvariablezandconsiderthespaceF(V)=F(V)¯0⊕F(V)¯1⊂EndV,V((z))
offieldsonV,givenbyF(V)p=(Xn∈Za(n)z−n−1pa(n)
=p,∀v∈V,a(n)v=0forn≫0),Herep(x)∈Z/2Zistheparityofx.Denoteby11∈F(V)¯0theidentityoperator,suchthat11(−1)=IdV,allothercoefficientsare0.Letıw,z(w−z)nandız,w(w−z)nbetwodifferentexpansionsof(w−z)nintoformalpowerseriesinthevariableswandz:ıw,z(w−z)n=Xi0(−1)n+iniwn−izi∈k[[w,w−1,z]],ız,w(w−z)n=Xi0(−1)iniwizn−i∈k[[w,z,z−1]].Ofcourse,ifn0thenıw,z(w−z)n=ız,w(w−z)n.Leta,b∈F(V).WesaythataislocaltobifthereissomeN∈Zsuchthata(w)b(z)ıw,z(w−z)N−(−1)p(a)p(b)b(z)a(w)ız,w(w−z)N=0.(1)TheminimalN=N(a,b)withthispropertyiscalledtheorderoflocalityofaandb.Intermsofthecoefficientslocalitymeansthatthefollowingidentitiesholdforallm,n∈Z:Xs0(−1)sNsa(m−s)b(n+s)−(−1)p(a)p(b)Xs6N(−1)sNN−sb(n+s)a(m−s)=0.(2)4MICHAELROITMANUsuallythelocalityisdefinedonlyforanonnegativeorderN[6,11],butwewillneedthispropertyinabiggergener