On Conformal Field Theory and Stochastic Loewner E

arXiv:hep-th/0308020v224Mar2004IHES/P/03/28Imperial/TP/2-03/30hep-th/0308020OnConformalFieldTheoryandStochasticLoewnerEvolutionR.FriedrichabandJ.KalkkinencaInstitutdesHautes´EtudesScientifiques,LeBois-Marie35,RoutedeChartres,Bures-sur-YvetteF–91440,FrancebUniversit´eParis-Sud,LaboratoiredeMath´ematiquesUniversit´eParisXI,F–91504Orsay,FrancecTheBlackettLaboratory,ImperialCollegePrinceConsortRoad,LondonSW72BZ,U.K.AbstractWedescribeStochasticLoewnerEvolutiononarbitraryRiemannsurfaceswithboundaryusingConformalFieldTheorymethods.WeproposeinparticularaCFTconstructionforaprobabilitymeasureon(clouded)paths,andcheckitagainstknownrestrictionproperties.Theprobabilitymeasurecanbethoughtofasasectionofthede-terminantbundleovermodulispacesofRiemannsurfaces.Loewnerevolutionshaveanaturaldescriptionintermsofrandomwalkinthemodulispace,andthestochasticdiffusionequationtranslatestotheVirasoroactionofacertainweight-twooperatoronauniformisedver-sionofthedeterminantbundle.PACS2003:02.50.Ey,05.50.+q,11.25.HfMSC2000:60D05,58J52,58J65,81T40Keywords:ProbabilityTheory;ConformalFieldTheoryEmail:[e-mail],[e-mail]1IntroductionThemotivationforthisarticlestemsfromtheworkofLawler,SchrammandWerner[1]inwhichtheyinvestigateonapurelyprobabilisticbasisthe“restrictionproperty”ofcertainprobabilityamplitudes.Thispropertycanbephrasedintermsofaclassofstochasticprocessesdefinedonpropersimplyconnecteddomainsofthecomplexplane,originallyintroducedinRef.[2].Theseprocesses–termedStochastic(orSchramm–)LoewnerEvo-lutions(SLE)–provedtobeofgreatvalueinprovidingarigorousbasisforcertainresultsinConformalFieldTheory(CFT)[3,4,5,6].ThepreciserelationofSLEstoCFThasremained,however,ratherspec-ulative.SomeaspectsofithavebeenclarifiedbuildingontherestrictionpropertyinRefs.[7,8],whereaninterpretationoftheparametersinvolvedinthe“Brownianbubbleprocess”asthecentralchargeandtheconformalweightofahighestweightrepresentationofanunderlyingCFTwasobtained.Thisledfurthertoanexplicitcalculationofthecorrelationfunctionsofthestress–energytensorinsertedattheboundaryfortrivialcentralchargec=0.Inthispaperwecontinuetoinvestigatethepropertiesoftheobjectsfoundin[1]andproposeapreciseCFTmodelforchordalSLEthattosomeextentgeneralisesit.Inaforthcomingpaper[9]wewillexpandonthemathemat-icalsideofthisarticle.Anintroductiontothepresentliteratureandtoremainingchallengesinparticularinthecaseofpercolationcanbefoundin[10,11].Recallthataclassofstatisticalmechanicsmodelsatthecriticaltemper-aturecanbedescribedintermsofCFT.Wewillinvestigatemodelsdefinedondomainswithboundaries,ormoregenerallyonsurfaceswithboundaries(openstringworld-sheets).Thechoiceofappropriateboundaryconditionswillgiverisetoalongchordaldomainwall,connectingtwopointsofthe(same)boundarycomponent.Sincethesefluctuatingcurvesexistonalllengthsscalesitismeaningfultoaskforthelimitingobject:Inthecaseofsimplyconnectedplanardomainsithasbeenshown[2]thatthescalinglimitcorrespondstoaclassofconformallyinvariantprobabilitymeasures,supportedbytheserandomcruves.Wewillshowherehowtorelate,inthegeneralcase,suchrandomcurvestothevacuumexpectationvaluesofcertainnonlocaloperatorsinsertedonthe(boundariesof)themanifold.Thepropertreatmentofrandomcurvesstartingfromapointinthebulkandconnectinganotherpointontheboundaryisrelatedtoorderdisorderlineswhichwewillnottreatinthepresentarticle.InthegeneralsituationwewillshowhowspecificdensitiesassociatedtochordalcurveshavethesamecovariancepropertiesunderconditioningasareknowninthecaseofchordalSLEontheupperhalf-plane.1Thediffusionequation,whichdescribesthestochasticprocess,canbethoughtofasarealSchr¨odingerequationofaquantummechanicalparticleontheboundary.TheHamiltonian,whichisasecond-orderhypo-ellipticdifferentialoperator,isthegeneratorofthestochasticprocess.Atzeroen-ergy,thatistosaywhentheHamiltonianannihilatesthewavefunction,theprobabilitydensityisamartingale.TheprobabilitydensitiesofSLEareconformalformsonthemodulispaceofRiemannsurfaceswithboundarycomponentsandonemarkedpointMg,b,1.TheycanbeassumedtobelongtohighestweightrepresentationsoftheVirasoroalgebraoftheCFT,theweightitselfbeingdeterminedbythevarianceofthedrivingBrownianmotioni.e.thediffusioncoefficient.FromthispointofviewthemartingalepropertycorrespondstorequiringthattheHamiltonianshiftstheconformalweightofthewavefunctionbytwo,andthattheresultingrepresentationisstillofhighestweight,andsingular,sothatitisequivalenttozerointhephysicalHilbertspace.Thefactthatrep-resentationsoftheVirasoroalgebrawereinvolvedinthedescriptionofSLEcurveswasnoticedalreadyin[7,12,8].AstheLoewnerprocesschangestheconformalclassoftheRiemannsur-face,italsogeneratesarandomwalkinthemodulispaceMg,b,1.Wedescribethisindetail,andwritedowntheprobabilitydensitiesassectionsofcertainlinebundlesonthisspace.Weshowinparticularthattheinducedmotiononthemodulispacehasatangentvectorfieldthatliesentirelyinarank-twovectorbundledeterminedbythesecond-orderdifferentialoperator.Supposewearegivenadifferentiablepathγtintheupperhalf-planeH.Thenwecanmakeapolygonalapproximationofitwith(infinitesimal)straightlines.TheRiemannmappin