A family of solvable non-rational conformal field

arXiv:0803.2099v2[hep-th]23May2008PreprinttypesetinJHEPstyle-HYPERVERSIONPTA/08-019Afamilyofsolvablenon-rationalconformalfieldtheoriesSylvainRibaultLaboratoiredePhysiqueTh´eoriqueetAstroparticules,UMR5207CNRS-UM2,Universit´eMontpellierII,PlaceE.Bataillon,34095MontpellierCedex05,[e-mail]:Wefindnon-rationalconformalfieldtheoriesintwodimensions,whicharesolvableduetotheircorrelatorsbeingrelatedtocorrelatorsofLiouvilletheory.Theirsymmetryalgebraconsistsofthedimension-twostress-energytensor,andtwodimension-onefields.Thetheoriescomeinafamilywithtwoparameters:thecentralchargecandacomplexnumberm.Thespecialcasem=0correspondstoLiouvilletheory(plustwofreebosons),andm=1correspondstotheH+3model.Inthecasem=2weshowthatthecorrelatorsobeythird-orderdifferentialequations,whichareassociatedtoasubsingularvectorofthesymmetryalgebra.Contents1.Introduction12.Anansatzforthecorrelationfunctions22.1ReminderoftheH+3-Liouvillerelation22.2Motivationoftheansatz43.Symmetryalgebraofthenewtheories53.1Lagrangiandescription53.2Symmetryalgebra74.Differentialequationinthecasem=294.1Third-orderBPZequation94.2ReformulationoftheBPZequationintermsofsymmetrygenerators114.3Subsingularvectorsofthesymmetryalgebra125.Concludingremarks131.IntroductionConformalfieldtheoriesintwodimensionshavemanyapplicationsinstringtheoryandcondensed-matterphysics.Insomeapplicationsthesystemunderconsideration,saystringsinagivenspace-time,dictatesthetheorytobestudied.Insomeothercaseshowever,aCFTisusedasaphenome-logicaltheoryforasystemwithoutaknownfirst-principlesdescription.TherelevanttheorymustthenbechosenamongtheknownCFTsaccordingtothedesiredpropertiesofthesystem.ThisapproachreliesontheexistenceofalargeenoughnumberofCFTsamongwhichtochoose,andonagoodenoughknowledgeoftheirproperties.ManyrationalCFTsareknown,andtheirpropertiesareoftenwell-understood.HowevermostCFTsarenon-rational,andonlyafewnon-rationalCFTshavebeensolvedsofar,amongthemLiouvilletheory[1]andtheH+3model[2].Itisthereforeinterestingtoexpandthesmallfamilyofsolvablenon-rationalCFTs.Theinspirationforthepresentarticlecomesfromtherelation[3]betweenthecorrelationfunctionsoftheH+3modelandcorrelationfunctionsofLiouvilletheory.Namely,correlatorsofgenericH+3primaryfieldsarerelatedtocorrelatorsinvolvingbothgenericLiouvilleprimaryfields,andtheparticularfieldV−12bwherebisrelatedtothecentralchargeofLiouvilletheorybyc=1+6(b+b−1)2.Thisfieldisknowntobeadegeneratefieldwithanullvectoratleveltwo,whichimpliesthatLiouvillecorrelatorsinvolvingthisfieldobeysecond-orderdifferentialequations[4].ButthereisaninfiniteseriesofdegeneratefieldsinLiouvilletheory,whichleadtohigher-orderdifferentialequations.ItisthereforenaturaltowonderwhetherLiouvillecorrelators–1–involvingsuchdegeneratefields–orevengenericfields–canberelatedtocorrelatorsinsomemodelswhichwouldbegeneralizationsoftheH+3model.Wewillproposeapositiveanswertothisquestion,andsketchsomepropertiesoftheresultingmodels.Incontrasttotheusualsituationinthestudyofconformalfieldtheory,westartwiththesolutionofthemodels,namelyanansatzfortheircorrelatorsintermsofLiouvillecorrelators.Theproblemisthentoreconstructthemodels,andinparticulartheirsymmetryalgebra,fromtheircorrelators.Thisisapriorianon-trivialproblem,becauseouransatzgivesusonlycorrelatorsofprimaryfields,butthenotionofaprimaryfieldisnotevendefinedifwedonotknowthesymmetryalgebra.Fortunately,thecorrectsymmetryalgebrawillbesuggestedbytheLagrangiandescriptionofthemodels,whichisverysimilartothefree-fielddescriptionoftheH+3model.(ThenewLagrangianshoweverdonotcorrespondtosigmamodels.)Wewillthencheckthatthissymmetryalgebraisconsistentwiththepropertiesofthecorrelators,inparticularinthecasewhentheyobeythird-orderdifferentialequations.2.AnansatzforthecorrelationfunctionsOuransatzwillbedirectlyinspiredbytheH+3-Liouvillerelation,wewillthereforebeginbyareminderofthisrelation.Moredetailsandreferencescanbefoundin[3].2.1ReminderoftheH+3-LiouvillerelationIntheconformalbootstrapapproach,theH+3modelisdefinedbyitssymmetryalgebra,andpri-maryfieldswhichtransforminacertainwaywithrespecttothisalgebra.Solvingthemodelmeansfindingthecorrelationfunctionsoftheprimaryfields;thesecorrelationfunctionsaresubjecttosymmetryandconsistencyconditions.Letusbrieflyreviewtheseobjects.Thechiralsymme-tryalgebraoftheH+3modelistheaffineLiealgebracsℓ2,whichisgeneratedbythreecurrentsJ3(z),J+(z),J−(z)withthefollowingoperatorproductexpansions(OPEs):J3(z)J3(w)=−k2(z−w)2+O(1),(2.1)J3(z)J±(w)=±J±(w)z−w+O(1),(2.2)J+(z)J−(w)=−k(z−w)2+2J3(w)z−w+O(1),(2.3)wherek2iscalledtheleveloftheaffineLiealgebra.Asaconsequence,theH+3modelisaconformalfieldtheorywithcentralchargec=3kk−2;thestress-energytensorwhichgeneratestheconformaltransformationscanbededucedfromthecurrentsbytheSugawaraconstruction[5]T=12(k−2)(J+J−)+(J−J+)−2(J3J3),(2.4)whereweusethefollowingdefinitionofthenormal-orderedproductofoperators:(AB)(z)=12πiIzdxdxx−zA(x)B(z).(2.5)–2–AffineprimaryoperatorsΦj(μ|w)withspinj,isospinμandworldsheetpositionwaredefinedbytheirOPEswiththecurrents,fromwhichtheirOPEswiththestress-energytensorfollows:J−(z)Φj