Null-vectors in Integrable Field Theory

arXiv:hep-th/9606068v219Feb1997SPhT/96/063LPTHE-96-23Null-vectorsinIntegrableFieldTheory.O.Babelona0,D.Bernardb0andF.A.Smirnova1aLaboratoiredePhysiqueTh´eoriqueetHautesEnergies2Universit´ePierreetMarieCurie,Tour161er´etage,4placeJussieu75252Pariscedex05-FrancebServicedePhysiqueTh´eoriquedeSaclay3F-91191,Gif-sur-Yvette,France.Abstract.Theformfactorbootstrapapproachallowstoconstructthespaceoflocalfieldsinthemassiverestrictedsine-Gordonmodel.Thisspacehastobeisomorphictothatofthecorrespondingminimalmodelofconformalfieldtheory.WedescribethesubspaceswhichcorrespondtotheVermamodulesofprimaryfieldsintermsofthecommutativealgebraoflocalintegralsofmotionandofafermion(Neveu-SchwarzorRamonddependingontheparticularprimaryfield).Thedescriptionofnull-vectorsreliesontherelationbetweenformfactorsanddeformedhyper-ellipticintegrals.Thenull-vectorscorrespondtothedeformedexactformsandtothedeformedRiemannbilinearidentity.Intheoperatorlanguage,thenull-vectorsarecreatedbytheactionoftwooperatorsQ(linearinthefermion)andC(quadraticinthefermion).Weshowthatbyfactorizingoutthenull-vectorsonegetsthespaceofoperatorswiththecorrectcharacter.Intheclassicallimit,usingtheoperatorsQandCweobtainanew,verycompact,descriptionoftheKdVhierarchy.WealsodiscussabeautifulrelationwiththemethodofWhitham.0MembreduCNRS1OnleavefromSteklovMathematicalInstitute,Fontanka27,St.Petersburg,191011,Russia2Laboratoireassoci´eauCNRS.3LaboratoiredelaDirectiondesSciencesdelaMati`ereduCommissariat`al’EnergieAtomique.11Introduction.Inthisarticlewepresentasynthesisoftheideasofthepapers[1]and[2].Inthefirstofthesepapersthespaceoffieldsforthesine-Gordonmodel(SG)wasdescribedintermsoftheformfactorspreviouslyobtainedinthebootstrapapproach[3].ThisdescriptionisbasedonratherspecialpropertiesofformfactorsfortheSGmodel.Namely,itusesthefactthattheformfactorswerewrittenintermsofdeformedhyper-ellipticdifferentials,allowingdeformationsofallthenicepropertiesoftheusualhyper-ellipticdifferentials:thenotionofdeformedexactformsandofthedeformedRiemannbilinearidentityareavailableforthem[4].Usingthesefactsithasbeenshownin[1]thatthesamenumberoflocaloperatorscanbeconstructedinthegenericcaseofthesine-Gordonmodelasatthefreefermionpoint.ThedeformedexactformsandthedeformedRiemannbilinearidentityarenecessaryinordertoreducethespaceoffieldstothepropersizebecauseinitsoriginalformfactordescriptionthespaceistoobig.Thedescriptionof[1]isbasicallyindependentofthecouplingconstant,butforrationalcouplingconstantthereisapossibilitytofindadditionaldegenerations.Theproblemwiththedescriptionofthespaceoffieldsobtainedin[1]isduetothefactthatitisdifficulttocompareitwiththedescriptioncomingfromConformalFieldTheory(CFT)orfromtheclassicaltheory.ThelattertwoarecloselyconnectedbecausetheVirasoroalgebracanbeconsideredasaquantizationofthesecondPoissonstructureofKdV.Inthedescriptionof[1],itisevendifficulttodistinguishthedescendentswithrespecttothetwochiralVirasoroalgebras.Ontheotherhandin[2]thesemi-classicallimitoftheformfactorformulaehasbeenunderstood.Thisopensthepossibilityofidentifyingallthelocaloperatorsbytheirclassicalanalogues.UsingthisresultwedecidedtotrytoconstructthemoduleofthedescendentsoftheprimaryfieldswithrespecttothechiralVirasoroalgebra.Theresultofthisstudyhappenedtobequiteinteresting.Letusformulatemorepreciselytheproblemsdiscussedinthispaper.Thesine-Gordonmodelisdescribedbytheaction:S=πγZ(∂μϕ)2+m2(cos(2ϕ)−1)
d2xwhereγisthecouplingconstant.Inthequantumtheory,therelevantcouplingconstantis:ξ=πγπ−γ.Thesine-Gordontheorycontainstwosubalgebrasoflocaloperatorswhich,asoperatoralgebras,aregeneratedbyexp(iϕ)andexp(−iϕ)respectively.Weshallconsideroneofthem,saytheonegeneratedbyexp(iϕ).Itisknownthatthissubalgebracanbeconsideredindependentlyoftherestoftheoperators,astheoperatoralgebraofthetheorywiththemodifiedenergy-momentumtensor:Tmodμν=TSGμν+iαǫμ,μ′ǫν,ν′∂μ′∂ν′ϕwhereα=πq6ξ(π+ξ).Thismodificationchangesthetraceoftheenergy-momentumtensorwhichisnow:Tmodμμ=m2exp(2iϕ).Thismodifiedenergy-momentumtensorcorrespondstotherestrictedsine-Gordontheory(RSG).Forrationalξπ,theRSGmodeldescribestheΦ[1,3]-perturbationsoftheminimalmodelsofCFT.InthispaperweconsideronlytheRSGmodel.Itisnaturalfromthephysicalpointofviewofintegrableperturbations[5]toexpectthatthespaceoffieldsfortheperturbedmodelisthesameasforitsconformallimit.ThelatterconsistsoftheprimaryfieldsandtheirdescendentswithrespecttothetwochiralVirasoroalgebras.Inthispaperweshallconsiderthedescendentswithrespecttooneofthesealgebras,thepossibilityofconsideringthedescendentswithrespecttotheotheroneisexplainedinSubsection2.2.TheVermamoduleoftheVirasoroalgebraisgeneratedbytheactionofthegeneratorsL−koftheVirasoroalgebraontheprimaryfield.Theirreduciblerepresentationcorrespondingtoagivenprimaryfieldisobtainedbyfactorizingoutthenull-vectors[6,7].OntheotherhandtheverypossibilityofintegrabledeformationsisduetothefactthatthereexistsacommutativesubalgebraoftheuniversalenvelopingalgebraoftheVirasoroalgebra:thealgebraoflocalintegrals[5,8,9].ThelocalintegralsI2k−1haveoddspins.LogicallyitmustbepossibletopresenttheVermamoduleasaresu