Free Field Realization of Vertex Operators for Lev

arXiv:math/9809078v1[math.QA]16Sep1998FREEFIELDREALIZATIONOFVERTEXOPERATORSFORLEVELTWOMODULESOFUqbsl(2)
YUJIHARAAbstract.FreefieldrealizationofvertexoperatorsforleveltwomodulesofUqbsl(2)
areshownthroughthefreefieldrealizationofthemodulesgivenbyIdzumiinRef.[4,5].WeconstructedtypesIandIIvertexoperatorswhenthespinoftheassociatedevaluationmoduleis1/2andtypeII’sforthespin1.1.IntroductionVertexoperatorsforthequantumaffinealgebraUqbsl(2)
haveplayedessentialrolesinthealgebraicanalysisofsolvablelatticemodelssincethepioneeringworksof[1,2,3].IntheseworkswhichanalyzetheXXZmodel,typeIvertexoperatorsareidentifiedwithhalfinfinitetransfermatricesastheirrepresentation-theoreticalcounterpartandtypeIIvertexoperatorsareinterpretedasparticlecreationoperators.Toperformconcretecomputationsuchasatraceofcompositionofvertexoperators,weneedfreefieldrealizationofmodulesandoperators.InthesaidexampleoftheXXZmodel,theintegralexpressionsofn-pointcorrelationfunctionswhicharespecialcasesofthetracesareobtainedthroughbosonizationoflevelonemoduleofUqbsl(2)
.Motivatedbytheseresults,Idzumi[4,5]constructedleveltwomodulesandtypeIvertexoperatorsaccompaniedbyspin1evaluationmodulesforUqbsl(2)
intermsofbosonsandfermionsandthencalculatedcorrelationfunctionsofaspin1analogueoftheXXZmodel.ThepurposeofthispaperistoextendIdzumi’sfreefieldrealizationtootherkindsofvertexoperatorsi.e.typeIandIIvertexoperatorsfortheleveltwomodulesassociatedwiththeevalutionmoduleofspin1/2andthetypeII’sforthespin1.TheresultsaregiveninSection3andtheirderivationisdiscussedinthefirstcaseinSection4.TheresultstogetherwithRef.[4,5]givethecompletesetofvertexoperatorsforleveltwomoduleofUqbsl(2)
andenableonetocalculateformfactorsofthespin1analogueoftheXXZmodel.RecentlyJimboandShiraishi[7]showedacoset-typeconstructionforthedeformedVirasoroalgebrawiththevertexoperatorsforUqbsl(2)
.TheyconstructedaprimaryoperatorforthedeformedVirasoroalgebraascosettypecompositionofvertexoperatorswhichmaybedenotedasUqbsl(2)
k⊕Uqbsl(2)
1/Uqbsl(2)
k+1.Wehopethatourresultswillbehelpful1forextendingthisworktothedeformedsupersymmetricVirasoroalgebrathroughUqbsl(2)
k⊕Uqbsl(2)
2/Uqbsl(2)
k+2.2.Freefieldrealizationofleveltwomodule2.1.Convention.InthefollowingwewilluseUtodenotethequantumaffinealgebraUqbsl(2)
.Unlessmentioned,wefollowthenotationsofRef.[4,5].Asforthefreefieldrepresentation,weslightlymodifytheconvention.ThequantumaffinealgebraUisanassociativealgebrawithunit1generatedbyei,fi(i=0,1),qh(h∈P∗)withrelationsq0=1,qhqh′=qh+h′,qheiq−h=qhh,αiiei,qhfiq−h=q−hh,αiifi,[ei,fi]∗=δijti−t−1iq−q−1,(ti=qhi)e3iej−[3]e2iejei+[3]eieje2i−eje3i=0,f3ifj−[3]f2ifjfi+[3]fifjf2i−fjf3i=0,whereP=ZΛ0+ZΛ1+ZδistheweightlatticeoftheaffineLiealgebrabsl(2)andP∗istheduallatticetoPwiththedualbasis{h0,h1,d}to{Λ0,Λ1,δ}withrespecttothenaturalpairingh,i:P×P∗→Z.WealsousecurrenttypegeneratorsintroducedbyDrinfeld[11][ak.al]=δk+l,0[2k]kγk−γ−kq−q−1,KakK−1=ak,Kx±kK−1=q±2x±k,[ak,x±l]=±[2k]kγ∓|k|/2x±k+l,x±k+lx±l−q±2x±lx±k+l=q±2x±kx±l+1−x±l+1x±k,[x+k,x−l]=γk−l2ψk+l−γl−k2φk+lq−q−1,whereψk,andϕkaredefinedasXk0ψkz−k=Kexp(q−q−1)Xk1akz−k ,∗[A,B]=AB-BA2Xk0φkzk=K−1exp−(q−q−1)Xk1a−kzk .Therelationbetweentwotypesofgeneratorsaret1=K,t0=γK−1,e1=x+0,e0t1=x−1,f1=x−0,t−11f1=x−10.ThehigestweightmoduleandtheevaluationmodulearedescribedcompactlyinRef.[4].Commutationandanticommutationrelationsofbosonsandfermionsaregivenby[am,an]=δm+n,0[2m]2m,{φm,φn}∗=δm+n,0ηm,ηm=q2m+q−2m.wherem,n∈Z+1/2or∈ZforNeveu-Schwarz-sectororRamond-sectorrespectively.FockspacesandvacuumvectorsaredenotedasFa,FφNS,FφRand|vaci,|NSi,|RiforthebosonandNSandRfermionrespectively.FermioncurrentsaredefinedasφNS(z)=Xn∈Z+12φNSnz−n,φR(z)=Xn∈ZφRnz−n.Q=Zαistherootlatticeofsl2andF[Q]bethegroupalgebra.Weuse∂as[∂,α]=2.2.2.V(2Λ0),V(2Λ1).ThehighestweightmoduleV(2Λ0)isidentifiedwiththeFockspaceF(0)+=Fa⊗(FφNSeven⊗F[2Q])⊕(FφNSodd⊗eαF[2Q]) ,(1)subscriptsevenandoddrepresentthenumberoffermions.Thehighestweightvectoris|vaci⊗|NSi⊗1.V(2Λ1)isF(0)−=Fa⊗(FφNSeven⊗eαF[2Q])⊕(FφNSodd⊗F[2Q]) (2)withthehighestweightvectorbeing|vaci⊗|NSi⊗eα.NotethatF(0)=F(0)−⊕F(0)+.F(0)=Fa⊗FφNS⊗F[Q].Theoperatorsarerealizedinthefollowingmanner.γ=q2,K=q∂,x±(z)=Xm∈Zx±mz−m=E±(z)E±(z)φNS(z)e±αz12±12∂,∗{A,B}=AB-BA3E±(z)=exp(±Xm0a−m[2m]q∓mzm),E±(z)=exp(∓Xm0am[2m]q∓mz−m),andd=−∂28+(λ,λ)4−∞Xm=1mNam−Xk0kNφNSk,(3)Nam=m[2m]2a−mam,NφNSk=1ηmφNS−mφNSm(m0),(4)wherethehigestweightvectorofthemoduleshouldbesubstitutedforλof(3).2.3.V(Λ0+Λ1).ThemoduleV(Λ0+Λ1)isidentifiedwithF(1)=Fa⊗FφR⊗eα2F[Q],(5)whereφR0|Ri=|Ri.Thehighestweightvectorisidentifiedwith|vaci⊗|Ri⊗eα2.OperatorsareconstructedinthesamewayasbeforeexceptthatsubscriptsforfermionsectorareRinsteadofNS.3.FreefieldrealizationsofvertexoperatorsLetV,V′beleveltwomodulesandV(k)zbeaspink/2evaluationmoduleofU.VertexoperatorswewillconsiderareU-linearmapsofthefollowingkinds[8,9]ΦV′,kV(z):V−→V′⊗V(k)z,(6)Ψk,V′V(z):V−→V(k)z⊗V′.(7)Vertexoperatorsoftheform(6,7)arecalledtypeIandIIrespectively.ComponentsofvertexoperatorsaredefinedasΦ(z)V′,kV=kXn=0Φn(z)⊗un,Ψ(z)k,V′V=kXn=0un⊗Ψn(z).3.1.typeIVertexOperatorsforlevel2andspin1/2.Wesho