The Two-Dimensional String as a Topological Field

arXiv:hep-th/9312190v124Dec1993TIFR/TH/93-61hep-th/9312190December1993TheTwo-dimensionalStringasaTopologicalFieldTheory∗SunilMukhiTataInstituteofFundamentalResearchHomiBhabhaRd,Bombay400005,IndiaABSTRACTAcertaintopologicalfieldtheoryisshowntobeequivalenttothecompactifiedc=1string.ThistheoryisdescribedinbothKazama-SuzukicosetandLandau-Ginzburgformulations.Thegenus-gpartitionfunctionandgenus-0multi-tachyoncorrelatorsofthec=1stringareshowntobecalculableinthisapproach.TheKPZformulationofnon-criticalstringtheoryhasanaturalrelationtothistopologicalmodel.∗TalkgivenattheNatoAdvancedResearchWorkshopon“NewDevelopmentsinStringTheory,ConformalModelsandTopologicalFieldTheory”,Cargese,May12-211993.BasedonworkdoneincollaborationwithC.Vafa[1]andwithD.Ghoshal[2].1IntroductionFormanyyearsitusedtobesaidthatstringtheoryisconsistentonlyin26spacetimedimensions.WiththeunderstandingthatanyCFTwithtotalcentralchargec=26isaconsistentbackgroundforbosonicstringpropagation,thisstatementhadtobemodified.Itremainedtruethatinsomesensethesimplestknownbackgroundforthebosonicstringwastheonewith26flat,noncompactspacetimedimensions.Therelevanceoftheconceptof“simplestknownbackground”stemsfromthefactthatwedonotyetfullyunderstandbackground-independentstringfieldtheory.Inthissituation,thetheoryinthesimplestbackgroundmayreasonablybeexpectedtofurnishcluesaboutthenatureofthefulltheory.Forexample,arbitrarybackgroundsinafieldtheorytendtospontaneouslybreaksomeoralloftheglobalsymmetries.Insuchback-groundswewouldbehardpressedtodiscoverthefullsymmetrystructureofthetheory.Thus,wemustlookforbackgroundswhichpreserveasmanysymmetriesaspossible–indeed,thisisprobablythemostreasonabledefinitionof“simplestbackground”.Withthisinmind,itbecomesclearthatthenoncompact26-dimensionalbackgroundisfarfrombeingthesimplestone.Indeed,justcompactifyingthespacetimeonatorusofappropriateshapecanproduce[3]largenumbersofKac-Moodycurrents,whoseintegralsareunbrokensymmetrygenerators.Butevenbetteristhebackgroundwithjust2spacetimedimensions,traditionallydescribedbyscalarfieldsX(z,¯z)andφ(z,¯z)ontheworld-sheet.TheunbrokensymmetriesofthistheorycorrespondtoinfinitelymanyholomorphiccurrentssatisfyingthewedgesubalgebraofW∞[4,5,6].Thisisthemostsymmetricknownphaseofthebosonicstring.Moreprecisely,themaximumnumberofunbrokensymmetriesarisewhenthefieldX(z,¯z)iscompactifiedonacircleofradius1√2,theso-calledself-dualradius.Thetwo-dimensionalstringbackground(alsoknownas“c=1stringtheory”)hasthefollowingenergy-momentumtensorforitsmattersector:T(z)=−12∂X∂X−12∂φ∂φ+√2∂2φ(1)whereX(z,¯z)andφ(z,¯z)areconventionallyknownasthe“matter”and“Liouville”fieldsrespectively.ThepresenceofatotalderivativeterminT(z)correspondstoanextratermintheworldsheetactionproportionaltoR√gR(2)φ.Thistermrendersthefunctionalintegralill-defined,andtostabilisethetheorywemustaddacosmologicaltermμR√gexp(αφ).ThismeansthatLiouvillemomentumisnotconserved.EverycorrelationfunctioncarriesapowerofμequaltominustheamountbywhichitviolatesLiouvillemomentumconservation,inmultiplesof√2.1Inparticular,itiseasytoseethatthepartitionfunctionofthec=1stringingenusghasthebehaviourZg(μ)∼μ2−2g(2)Indeed,forthetwo-dimensionalstringcompactifiedontheself-dualradius,thepartitionfunctionisknownexplicitlyfrommatrixmodels[7]andisgivenbyZself−dualg(μ)=μ2−2gχg,χg=B2g2g(2g−2)(3)whereχgisthevirtualEulercharacteristicofthemodulispaceofgenus-gRiemannsurfaces,whichisproportionaltotheBernoullinumberB2g.Ithaslongbeenachallengetoreproducethisresultfromacontinuumformulationoftwo-dimensionalstringtheory.Othermatrix-modelresultsforthistheory,partic-ularlythecorrelationfunctionsof“tachyons”ingenus0,havebeenre-derivedinthecontinuumonlyafterconsiderabledifficultyandafterresortingtosomewhatarbitrarycontinuationsoftheparametersofthetheory[8,9,10,11,12].Oneofthecrucialdifficultiesincontinuumcalculationsisthatthetheoryisnotperturbativeinthecos-mologicalconstantμ(asisevidentfromEq.2).AnotherproblemisthatstandardCFTtechniquesproducetheamplitudesforlocalvertexoperatorsinsertedatfixedpointsontheworldsheet,whichmustthenbeintegratedovertheworldsheettogivethephysicalamplitudes.Inatheorycoupledtogravity,physicalanswersforcorrelationsofinte-gratedoperators(whicharetypicallyquitesimple)shouldnothavetodependonfirstobtainingthelocalanswerandexplicitlyperformingacomplicatedintegral.InwhatfollowsIwillshowthatthereisatopologicalfieldtheorywhichisequivalenttoc=1stringtheoryattheself-dualradius,andthatthemanifestlytopologicalformu-lationpermitsthecomputationofthepartitionfunctionandamplitudesinanygenus,andatnonzerocosmologicalconstant,withoutresortingtoanalyticcontinuations.Thismeansthatwehaveacontinuumformulationofthisstringtheorywhichisapparentlyaspowerfulastheverysuccessfulmatrix-modeldescription.Thisdiscoverymaylenditselftogeneralizationandteachussomethingaboutstringtheoryitselfratherthanaboutsomespecificbackground,unlikethematrixmodelswhichhavesofarresistedattemptstogobeyondthespecificbackgroundswhichtheyrepresent.2TopologicalSymmetryofStringBackgroundsTopologicalsymmetrywasthoughttobeaspecialpropertyofsomestringbackgrounds.Mor