Conformal Field Theory Techniques in Random Matrix

arXiv:hep-th/9907060v18Jul1999.SPHT/t98/112ConformalFieldTheoryTechniquesinRandomMatrixmodelsIvanK.Kostov∗†C.E.A.-Saclay,ServicedePhysiqueTh´eoriqueF-91191Gif-sur-Yvette,FranceInthesenotesweexplainhowtheCFTdescriptionofrandommatrixmodelscanbeusedtoperformactualcalculations.Ourbasicexampleisthehermitianmatrixmodel,reformulatedasaconformalinvarianttheoryoffreefermions.WegiveanexplicitoperatorconstructionofthecorrespondingcollectivefieldtheoryintermsofabosonicfieldonahyperellipticRiemannsurface,withspecialoperatorsassociatedwiththebranchpoints.Thequasiclassicalexpressionsforthespectralkernelandthejointeigenvalueprobabilitiesaretheneasilyobtainedascorrelationfunctionsofcurrent,fermionicandtwistoperators.Theresultforthespectralkernelisvalidbothinmacroscopicandmicroscopicscales.Attheendwebrieflyconsidergeneralizationsindifferentdirections.BasedonthetalkoftheauthorattheThirdClaudeItzykson-Meeting,”IntegrableModelsandApplicationstoStatisticalMechanics”,Paris,July27-29,1998,andattheworkshop“Randommatricesandintegrablesystems”,Univ.ofWarwick,November2-41998.July1999∗MemberofCNRS†[e-mail]rixmodelshavevariousapplicationsinratherdifferentdomains,andsometimeslanguagebarrierspreventstheflowofideasandknowledgefromonefieldtoanother.Forexample,suchpowerfultechniquesastheconformalfieldtheory(CFT)descriptionoftherandommatrixmodelsandtheirrelationwiththeintegrablehierarchies,whichweredeveloppedextensivelybythestringtheoristsintheearly90’s,arepracticallyunknowntothemesoscopicphysicists.ThislectureisanattempttoexplaintheusesofthetheCFTdescriptioninalanguage,whichisacceptablebybothcommunities.Wethereforeavoidedany“physical”interpretationandconcentratedonthemethodassuch.Theonlythingthereaderissupposedtoknowarethebasicsofthetwo-dimensionalconformalfieldtheory.⋄⋄⋄Thestatisticalensemblesofrandommatricesoflargesize(matrixmodels)havebeenintroducedin1951byWignerinordertoanalyzethespectralpropertiesofcomplicatedsystemswithchaoticbehavior[1].InthisapproachtheHamiltonianofachaoticsystemisconsideredasalargematrixwithrandomentries.Consequently,theanalyticalstudiesofrandommatrixensemblescarriedoutinthenext25years(seetheMehta’sbook[2])wereorientedtothecalculationofthespectralcorrelationfunctionsorjointeigenvalueprobabilities.IfMisanN×Nhermitianrandommatrix,thenthespectralcorrelationfunctionρ(λ1,...,λn)isdefinedastheprobabilitydensitythatλ1,...,λnareeigenvaluesofMρ(λ1,...,λn)=(N−n)!N!*nYi=1δ(M−λi)+.(1.1)AllspectralcorrelationfunctionsareexpressedasdeterminantsofasinglekernelK(λ,μ)(thespectralkernel).Thespectralkernelcanbeevaluatedbythemethodoforthogonalpolynomials[2].ItslargeNasymptoticsischaracterizedbytheinterpositionofasmoothbehaviorandfastoscillationswithwavelength∼1/N(inascalewherethetotalrangeofthespectrumiskeptfinite).Thesmoothlargedistancebehaviordependsontheconcreteformofthematrixpotential.Onthecontrary,themicroscopicbehaviorcharacterizedbyoscillationsdependsonlyonthesymmetrygroup(theunitarygroupinthecase)andfallintoseveraluniversalityclasses.Itisthemicroscopicbehaviorofthespectralcorrelations,whichisinterestingfromthepointofviewofapplicationstochaoticsystems.Areviewofthelatestdevelopmentsinthisdirection,inparticulartheserelatedtotherecentmonteCarlodatainlatticeQCD,canbefoundin[3].⋄⋄⋄Thediscoveryby‘tHooftofthe1/Nexpansion[4]gaveameaningofthesmoothpartofthespectralcorrelatorsandopenedthepossibilityofusingrandommatrixmodelstosolvevariouscombinatorialproblems,thesimplestofwhichistheenumerationofplanargraphs[5,6].Herewecanmentiontheexactsolutionofvariousstatisticalmodelsdefinedonrandomsurfaces[7,8,9],thematrixformulationofthe2dquantumgravity[10,11]andsomemoredifficultcombinatorialproblemsastheenumerationofthe”duallyweighted”planargraphs[12]andthebranchedcoveringsofacompactRiemannsurface[13].1Inthiskindofproblemsthesolutionisencodedinthe1/Nexpansionoftheloopcorrelationfunctions,whicharethecorrelationfunctionsofthecollectivefieldvariableW(z)=tr1z−M.(1.2)InthelargeNlimitthecorrelationfunctionsoftheresolvent(1.2)aremeromorphicfunc-tionswithcutsalongtheintervalswherethespectraldensityisnonzero.Thediscontinuityalongthecutsgivesthesmoothpartofthejointeigenvalueprobabilities(1.1).Indeed,wehavetheevidentrelationhW(z1)...W(zn)ic=Zdλ1...dλn(z1−λ1)...(zn−λn)ρ(λ1,...,λn)(1.3)wherehicmeansconnectedcorrelator.ThefirstexactresultsinthelargeNlimitwereobtainedbydirectapplicationofthesaddlepointmethod[5],butlateritwasrecognizedthatamorepowerfulmethodisprovidedbythesocalledloopequations[14],whoseiterativesolutionallowsonetoreconstructorderbyorderthe1/Nexpansion.Themostefficientiterativeprocedureprovedtobethe”moment’sdescription”[15],whichallowedtocalculatethefreeenergyandloopcorrelatorsforanarbitrarypotentialupto1/N4terms.⋄⋄⋄Intheearly90’s,afterthepublicationoftheseminalpapers[11],thetworesolu-tiontechniquesinrandommatrixmodels(orthogonalpolynomialsandloopequations)developedrapidlyandwererecognizedasparticularcasesofwelldevelopedmathematicalmethods.Themethodoforthogonalpolynomialswasreformulatedintermsofthetheoryofτfunctionsofintegra