Topological Matrix Models, Liouville Matrix Model

arXiv:hep-th/0310287v528Jul2005PreprinttypesetinJHEPstyle-HYPERVERSIONhep-th/0310287TIFR/TH/03-21TopologicalMatrixModels,LiouvilleMatrixModelandc=1StringTheory∗SunilMukhi†TataInstituteofFundamentalResearch,HomiBhabhaRd,Mumbai400005,IndiaAbstract:ThisisareviewofsomebeautifulmatrixmodelsrelatedtothemodulispaceofRiemannsurfacesaswellastononcriticalc=1stringtheoryatself-dualradius.TheseincludethePennermodelandtheW∞model,whichhavedifferentoriginsbutareequivalenttoeachother.Inthefinalsection,whichisnewmaterial,itisshownthatthesemodelsarealsoequivalenttoaLiouvillematrixmodel.WespeculatethatthismightbeinterpretedintermsofND-instantonsofthec=1string.Keywords:Stringtheory.∗BasedonlecturesdeliveredattheIPMWorkshoponStringTheory,Bandar-e-Anzali,Iran,inOctober2003.†Email:[e-mail]12.c=1atR=152.1Continuumformulationofc=152.2Matrixformulationofc=162.3FreeEnergy82.4Tachyoncorrelators93.RiemannSurfacesandthePennerMatrixModel113.1ModulispaceofRiemannsurfacesanditstopology113.2Quadraticdifferentialsandfatgraphs123.3ThePennermodel163.4Double-scaledPennermodel184.TheW∞MatrixModel194.1c=1amplitudesandW∞194.2RelationtothePennermodel224.3RelationtotheKontsevichmodel235.LiouvilleMatrixModelandD-Instantons245.1UnperturbedW∞andtheLiouvilleMatrixModel255.2InterpretationoftheModel255.3TachyonPerturbations276.Conclusions291.IntroductionInrecentmonthstherehasbeenarevivalofinterestinthenoncriticalc=1bosonicstring[1,2,3](anditsworldsheetsupersymmetriccounterparts[4,5,6]).Newinsighthasbeengainedintothisrelativelysimplestringtheory,usingseveraldevelopmentsthatcameafterthepreviousmatrixrevolution:D-branes[7],M(atrix)Theory[8,9],AdS/CFT[10]andtheunderstandingofboundarystatesinLiouvilletheory[11,12,13].–1–Mostoftherecentworkhascentredonmatrixquantummechanics,whosedouble-scaledlimitisbelievedtorepresentthec=1string(atleastperturbatively).Thebasicideaisthatthematrixofthismodelisthematrix-valuedtachyonontheworldlineofacollectionofND0-branes.Therearealsoclaimsthatanonperturbativelyconsistentversionofmatrixquan-tummechanicscanbeformulated.Thisissupposedtobeequivalentnottothebosonicc=1string,buttoaworldsheetsupersymmetricversion,theˆc=1type0Bstring[4,5].Wewillnotdiscussthislattertheoryhere.Butmanyoftheobservationsinthepresentreviewpresumablycan,andshould,begeneralisedtothenoncriticaltype0B(and0A)stringbackground.Initssimplest(un-orbifolded)form,thec=1stringhasatranslationallyinvari-antspace/timedirectionX.IfitisEuclidean,itcanbecompactifiedonacircleofradiusR.Inthatcase,allphysicalquantities(partitionfunctionandamplitudes)de-pendonR.TheEuclideantheorycanbeinterpretedasafinite-temperaturetheorywithRlabellingtheinversetemperature.ThevalueR=1(inunitswhereα′=1)isspecialbecausethec=1CFTisthenself-dualunderR→1R(1.1)Inthisarticlewewillfocusonthisbackground:theEuclideanc=1stringwiththeXdirectioncompactifiedattheself-dualradius.Forshort,wewillrefertothetheoryas“c=1,R=1”.Therearemanyindicationsthatstringtheoryistopologicalinthisbackground.Theterm“topologicalstringtheory”isusuallytakentomeanastringtheorywherethemattersectorhasatwistedN=2superconformalalgebrawithcentralchargezero.Tomakeitastringtheory,thismatterthenhastobecoupledtotopologicalworldsheetgravity[14,15],whichcanbedescribedbyasetofbosonicandfermionicghostswithtotalcentralchargezero.ThefermionicchargeofthetwistedN=2algebraistheBRSTchargedefiningphysicalstates.AclassicexampleofsuchatheoryisasuperstringonaCalabi-Yaubackground,whosesuperconformalworldsheettheoryhasN=2supersymmetrywithcentralchargec=9.Onmakingthestandardtopologicaltwistofthesuperconformalalgebra:T(z)→T(z)+12∂J(z)(1.2)whereJ(z)istheU(1)current,thealgebraacquirescentralcharge0andwehaveasuitablemattersystemforatopologicalstring[16].Itisknownthatordinarystringtheory,includingtheghostsector,canbeconsid-eredastopologicalmatter[17,18,19],whichmakesthedistinctionbetweentopologi-calandnon-topologicaltheorieslessclear-cut.Evencriticalbosonicandsuperstringsareknowntobe“topological”inthissense.Thebestdistinctiononecanmakeis–2–thatthelattertheoriesare“alreadytwisted”whiletheconventionaltopologicalthe-oriesareformulatedasN=2superconformaltheoriesandthengivenatopologicaltwist.TopologicalstringtheoriesaregenerallyrelatedtothetopologyofthemodulispaceofRiemannsurfaces.Forexample,“pure”topologicalgravitydescribesinter-sectiontheoryoncohomologyclassesassociatedtomodulispace,ortovectorbundlesonmodulispace[15].Someoftheindicationsthatc=1,R=1istopologicalarisefromitsrelationtoothertheories.Thepartitionfunctionofc=1,R=1iscloselyrelatedtothatofthePennermatrixmodel[20,21],amodelconstructedtocounttheEulercharacteristicofthemodulispaceofpuncturedRiemannsurfaces.Amplitudesinc=1,R=1aresummarisedintheformofW∞constraints[22]andthepartitionfunctionoftheperturbedtheoryisaτ-functionofanintegrablehierarchy1.Thisτ-functioninturncanbewrittenasamatrixmodel,theW∞matrixmodel[24](thishasbeenpreviouslyreferredtoasthe“Kontsevich-Pennermodel”[22]anda“Kontsevich-typemodelforc=1”[24]).Anotherindicationofthetopologicalnatureofc=1,R=1isthatitisdualtothetopological2