1006.1752v1The vertex algebra M(1)^+ and certain a

arXiv:1006.1752v1[math.QA]9Jun2010THEVERTEXALGEBRAM(1)+ANDCERTAINAFFINEVERTEXALGEBRASOFLEVEL−1DRAˇZENADAMOVI´CANDOZRENPERˇSEAbstract.Wegiveacosetrealizationofthevertexoperatoralge-braM(1)+withcentralchargeℓ.WerealizeM(1)+asacommu-tantofcertainaffinevertexalgebrasoflevel−1inthevertexalgebraLC(1)ℓ(−12Λ0)⊗LC(1)ℓ(−12Λ0).WeshowthatthesimplevertexalgebraLC(1)ℓ(−Λ0)canbe(conformally)embeddedintoLA(1)2ℓ−1(−Λ0)andfindthecorrespondingdecomposition.Wealsostudycertaincosetsubalge-brasinsideLC(1)ℓ(−Λ0).1.IntroductionInthelastfewyearsvarioustypesofW–algebrashavebeenstudiedintheframeworkofvertexoperatoralgebras(see[Ar],[AM2],[AM3],[DLY],[KW2]).InthispaperwewillbefocusedonW–algebraswhichadmitcosetrealization.ToanyvertexalgebraVanditssubalgebraU,onecanassociateanewvertexalgebraCom(U,V)={v∈V|unv=0forallu∈U,n≥0}calledthecommutant(orcoset)ofUinV.Thisisaveryimportantcon-structioninthetheoryofvertexoperatoralgebras,becauseitgivesareal-izationofalargefamilyofW–algebras.Anotherimportantconstructionistheorbifoldconstruction,whereanewvertexoperatoralgebraisobtainedasinvariantsinagivenvertexoperatoralgebrawithrespecttothefiniteautomorphismgroup.Asweshallseeinourpaper,somevertexalgebrasadmitbothrealizations,cosetandorbifold.InthispaperweconsidercertaincosetvertexalgebrasforvertexalgebrasassociatedtoaffineLiealgebras.LetgbeasimpleLiealgebraoftypeXn,ˆgtheassociatedaffineLiealgebraoftypeX(1)n,andLX(1)n(kΛ0)thesimplevertexoperatoralgebraassociatedtoˆgoflevelk∈C,k6=−h∨.Fork,m∈Z0,LX(1)n((k+m)Λ0)isasubalgebraofLX(1)n(kΛ0)⊗LX(1)n(mΛ0),andonehastheassociatedcosetvertexoperatoralgebraCom(LX(1)n((k+m)Λ0),LX(1)n(kΛ0)⊗LX(1)n(mΛ0)).(1.1)2000MathematicsSubjectClassification.Primary17B69,Secondary17B67,17B68,81R10.12DRAˇZENADAMOVI´CANDOZRENPERˇSEAlthoughtherearenoprecisegeneralresults(tothebestofourknowledge)aboutthestructureofthesecosets,itisbelievedthatthesevertexoperatoralgebrasarefinitelygeneratedandrational.In[AP2],wepresentavertex-algebraicproofofthefactthatinthecasek=1andaffineLiealgebrasoftypesD(1)nandB(1)n,thecoset(1.1)isisomorphictotherationalvertexoperatoralgebraV+L.Thesituationisevenmorecomplicatedforgeneralk,m∈C,suchthatk,m,k+m6=−h∨.ThenonehasthecosetvertexoperatoralgebraCom(eLX(1)n((k+m)Λ0),LX(1)n(kΛ0)⊗LX(1)n(mΛ0)),(1.2)whereeLX(1)n((k+m)Λ0)isacertainaffinevertexoperatoralgebraassociatedtoX(1)noflevelk+m(notnecessarilysimple).Inthispaperweidentifysomespecialcasesofsuchcosets,anditturnsoutthattheyarenotrational.Theconstructionin[AP2]isbasedonfermionicconstructionofvertexalgebrasandcertainconformalembeddings.Inthepresentpaperweusebosonicconstructionofvertexalgebrasandconstructnewconformalem-beddingsofaffinevertexalgebrasatlevel−1.ByapplyingthebosonicrealizationoftheaffinevertexalgebrasLA(1)1(−12Λ0)andLC(1)ℓ(−12Λ0)(cf.[FF])weconsidercosetvertexalgebrasCom(LA(1)1(−Λ0),LA(1)1(−12Λ0)⊗LA(1)1(−12Λ0))and(1.3)Com(eLC(1)ℓ(−Λ0),LC(1)ℓ(−12Λ0)⊗LC(1)ℓ(−12Λ0)).Itisinterestingthatthesecosetshavecentralcharge1.WeshowthatthesecosetsareisomorphictoM(1)+,whereM(1)istheHeisenbergvertexoperatoralgebraofrank1,andM(1)+istheZ2–orbifoldvertexalgebrastudiedin[DN1].ThestructuretheoryofM(1)+showsthatthesecosetsareirrationalvertexoperatoralgebrasandisomorphictoW(2,4)–algebrawithcentralchargec=1.Bycombiningresultsfrom[A2]andthepresentpaper,weclassifyirre-ducibleordinaryeLC(1)ℓ(−Λ0)–modules.Webelievethatthe(tensor)cate-goryofeLC(1)ℓ(−Λ0)–modulesisrelatedtothe(tensor)categoryofM(1)+–modules.Weplantoaddressthiscorrespondenceinourforthcomingpubli-cations.Generalizing(1.3),weuseanaturalrealizationofthevertexoperatoralgebraLA(1)1(−Λ0)⊗ℓasasubalgebraofLC(1)ℓ(−12Λ0)⊗LC(1)ℓ(−12Λ0),andprovethatCom(LA(1)1(−Λ0)⊗ℓ,LC(1)ℓ(−12Λ0)⊗LC(1)ℓ(−12Λ0))isisomorphictoM(1)+,whereM(1)istheHeisenbergvertexoperatoral-gebraofrankℓ.3Ourconstructionisbasedonanew,interestingconformalembeddingofaffinevertexoperatoralgebrasatlevel−1whichcanbeofindependentin-terest.WeshowthatLC(1)ℓ(−Λ0)isconformallyembeddedintoLA(1)2ℓ−1(−Λ0)andthatLA(1)2ℓ−1(−Λ0)∼=LC(1)ℓ(−Λ0)⊕LC(1)ℓ(−2Λ0+Λ2),whichimpliesthatLC(1)ℓ(−Λ0)isZ2–orbifoldofLA(1)2ℓ−1(−Λ0).ByusingconformalembeddingswestudycertaincategoriesofA(1)2ℓ−1–modulesoflevel−1from[AP1]asaC(1)ℓ–modules.ItturnsoutthatirreduciblehighestweightA(1)2ℓ−1–modulesLA(1)2ℓ−1(−(n+1)Λ0+nΛ1)andLA(1)2ℓ−1(−(n+1)Λ0+nΛ2ℓ−1)(n∈Z0)arealsoirreducibleasC(1)ℓ–modules.Thisresultisanaffineanalogueoftheisomorphismoffinite-dimensionalCℓ–modules:VA2ℓ−1(nω2ℓ−1)∼=VA2ℓ−1(nω1)∼=VCℓ(nω1).UsingtheseconformalembeddingswealsoshowthatthecosetCom(LA(1)1(−Λ0)⊗ℓ,LC(1)ℓ(−Λ0))isisomorphictoM(1)+,whereM(1)istheHeisenbergvertexoperatoral-gebraofrankℓ−1.2.PreliminariesLetVbeavertexalgebra([B],[FHL],[FLM],[LL]).ForasubalgebraUofVdenotebyCom(U,V)={v∈V|unv=0forallu∈U,n≥0}(2.4)thecommutantofUinV(cf.[FZ],[GKO],[LL]).Then,Com(U,V)isasubalgebraofV(alsocalledcosetvertexalgebra).Lethbeafinite-dimensionalvectorspaceequippedwithanondegeneratesymmetricbilinearformh·,·i,consideredasanabelianLiealgebra.Letˆh=h⊗C[t,t−1]⊕CKbeitsaffinizationwiththecenterK.ThenthefreebosonicFockspaceM(1)=S(h⊗t−1C[t−1])isasimplevertexoperatoralgebraofcentralchargeℓ=dimh,withVi