loop equations for the d-dimensional n-hermitian matrix model

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1、LoopEquationsforthed-dimensionaln-HermitianMatrixmodelJorgeAlfaroFacultaddeF´ısicaUniversidadCat´olicadeChileCasilla306,Santiago22,Chilejalfaro@lascar.puc.clNovember81993AbstractWederivetheloopequationsforthed-dimensionaln-Hermitianmatrixmodel.Theseareaconseq

2、uenceoftheSchwinger-Dysonequa-tionsofthemodel.MoreoverweshowthatinleadingorderoflargeNtheloopequationsformaclosedset.Inparticularwederivetheloopequationsforthen=2kmatrixmodelrecentlyproposedtode-scribethecouplingofTwo-dimensionalquantumgravitytoconformalmatte

3、rwithc>1.arXiv:hep-th/9311086v116Nov1993PUC-FIS28-93November199301IntroductionThelastthreeyearshaveseenarapiddevelopmentintheknowledgerelativetotwo-dimensionalquantumgravitycoupledtoconformalmatterwithc≤1.Thishasbeenpossiblebytheuseofmatrixmodeltechniquesandt

4、helargeNexpansion,combinedwiththedoublescalinglimit[1]Veryrecentlyanewinterpretationofthedoublescalinglimitasafinitesizerescalinghasemerged[2][3]basedontherenormalizationgroupappliedtolargeNmodels[4][5].Althoughthenewtechniqueshaveprovidedaverydeepunderstandin

5、goftwo-dimensionalquantumgravitycoupledtoc≤1conformalmatter,abarrierhasemergedatc=1,avoidingthecontinuationofthepreviousresultsbeyondthispoint.Thesamebarrierappearsinthecontinuumtreatmentofnon-criticalstrings.[6]Variousapproachestosurpassthisproblemhavebeensu

6、ggested.Amongthemwementiontheuseofreducedmodelstodiscussthed-dimensionalmatrixmodel[7]whichistherightmodeltodiscussthed-dimensionalstring,VeryrecentlywehaveobtainedtheloopequationsforthismodelforarbitraryNandshowthattheyformaclosedsetinleadingorderoflargeN[8]

7、.Forthemomenttheyhavenotbeensolvedbutweseethemasanimportantexactresultthatmaypermittocrossthebarrieratc=1.Lastyeara2n-Hermitianmatrixmodelinzerodimensionshavebeenproposedtodescribeasystemwithc=n/2coupledto2dgravity[9].Soasolutionofthismodelisanotherwaytocross

8、thebarrier.Fouryearsagoweobtainedtheloopequationsforthezerodimensionaltwo-matrixmodel[10],extendingthesolutionofMethaforthepartitionfunction[11].Lastyearwewereabletosolvetheseequationscom