Insection7wetryto ndacontinuumdescriptionoftheholestatesofthematrixmodelwiththehelpoftheconservedchargesofthetheory.Fromthedescriptionoftheconservedchargesinthefermionicformulationofthematrixmodelitiseasytoevaluatetheseconservedchargesforagivenholestate.Thesearegivenineq.(7.1).Usingtheknownrelationbetweentheconservedchargesinthematrixmodelandthose(Qj,m’s)inthecontinuumtheory,wecalculateineq.(7.2)theexpectedQj,mfortheholestates.Whateverbethecontinuumdescriptionoftheholemustcarrythesamecharges.Ontheotherhandouranalysisofsection3givesanexplicitexpressionfortheQj,m’sin
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Two dimensional string theory is known to have an infinite dimensional symmetry, both in the continuum formalism as well as in the matrix model formalism. We develop a systematic procedure for computing the conserved charges associated with these symmetrie
termsoftheboundarystateoftheD-brane.Thisgivesconstraintsontheboundarystatedescribingaholestate,butdoesnotdetermineitcompletely.Theproposalsmadein[5,29]satisfytheseconstraints;howeverwearguethattheydonotreproducetheexpectedclosedstringpro leofaholestate.Weproposeapossiblecandidatefortheseholsstatesbasedonsomequalitativearguments,butthereisnode niteconclusionyet.EarlierworkontherelationbetweensymmetriesofthematrixmodelandthoseinthecontinuumdescriptionwascarriedoutinthelimitofzeropotentialfortheLiouville eld,andwasbasedonthecomparisonofthesymmetryalgebrasinthetwodescriptions.Inordertocompareourresultstotheearlierresults,westudyinsection8thelimitofourresultsaswetakethepotentialfortheLiouville eldtozero.Weshow rstofallthatthislimitexistsandgivesawellde nedrelation(8.11)betweentheconservedchargesinthecontinuumtheoryandthoseinthematrixmodel.Furthermore,uptonormalizationfactorstheserelationsareconsistentwiththeearlierresultsobtainedbyworkingdirectlywiththeLiouvilletheorywithoutanypotentialterm.WealsostudythelimitofthediscretestateclosedstringbackgroundproducedbytheD0-braneboundarystateaswetaketheLiouvillepotentialtozerokeepingtheenergyoftheD0-brane xed.We ndthattheclosedstring eldcon gurationapproachesa nitevalue(8.14)inthislimit.Theanalysisofsection3expressestheconservedchargescarriedbyaD-braneintermsofitsboundarystate.Thisrelationisspeci ctoD-branessinceonlyD-branesaredescribedbyboundarystates.Insection9wemanipulatetheresultsofsection3torewritetheconservedchargeintermsoftheasymptoticvaluesofcertaincomponentsofclosedstring eldsproducedbythebraneforlargenegative .The nalformula,givenin(9.10),istheanalogoftheGausslawforelectrodynamicsrelatingtheelectricchargetoasymptoticelectric eld,ortheADMformulaingravityrelatingthemassofasystemtotheasymptoticgravitational eld.Theserelationsbetweenconservedchargesandasymptoticclosedstring eldsareexpectedtoholdforanysystemintwodimensionalstringtheory,evenifthesystemcannotberegardedasacollectionofD-branes.
Insection10weusethisresulttodeterminetheconservedchargescarriedbytheblackholesolutionofthetwodimensionalstringtheory.ForsimplicitytheanalysisofthissectioniscarriedoutforvanishingpotentialfortheLiouville eld,whichinthematrixmodeldescriptioncorrespondstothefermilevelcoincidingwiththemaximumofthepotential.Althoughtheblackholewasinitiallyconstructedasasolutionofthee ective eldtheory[30]orasanexactconformal eldtheory[31],itispossibletorepresentitasasolutioninstring eldtheorybyusinganiterativeprocedureforsolvingtheequationsofmotionofstring eldtheory[49].Usingthiswecan ndtheasymptoticclosedstring eldcon gurationassociatedwiththeblackholeandhencethechargesQj,mcarriedbytheblackhole.We ndthattheblackholecarriesonlytheconservedchargeQ1,0;all
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Two dimensional string theory is known to have an infinite dimensional symmetry, both in the continuum formalism as well as in the matrix model formalism. We develop a systematic procedure for computing the conserved charges associated with these symmetrie
otherchargesQj,mfor(j,m)=(1,0)ingtheknownrelation(6.6)betweenQj,mandtheconservedchargesinthematrixmodeldescriptionofthesystemwecanthenconstrainthepossiblecon gurationsinthematrixmodelwhicharecompatiblewiththeseconservedcharges.Inparticularwe ndthatthematrixmodeldescriptionoftheblackholemustconsistofalargenumberoflowenergyfermion-holepairsinsteadofa nitenumberof niteenergyfermionsandholes.
Thisdescriptionoftheblackholeposesanapparentpuzzle.Sinceinthematrixmodeldescriptionthefermionsarenon-interacting,andsincetheblackholebackgrounddi ersfromtheusualvacuumintermsofcreationofalargenumberoflowenergyfermion-holepairs,aclassicalD0-branecarrying niteenergyshouldnotbeabletorecognizethedi erencebetweenablackholeandtheordinaryvacuum.Canthisbetrueintwodimensionalstringtheory?WhileacompleteanswertothisquestionrequiresstudyingtheD0-branemotioninthesebackgroundstoallordersinα′,weshowthatatleastintheapproximationwherewetaketheD0-braneworld-lineactiontobeoftheDirac-Born-Infeldform,theclassicalD0-branecannotdistinguishtheblackholefromtheusuallineardilatonbackground.Thisisshownbydemonstratingthatthereisacoordinatetransformationthatconvertsthee ectivemetricseenbytheD0-braneintheblackholebackgroundtothee ectivemetricseenbytheD0-branetothelineardilatonbackground.Thiscoordinatetransformationactsonlyonthespacecoordinateandleavesthetimecoordinateunchanged.Thisisconsistentwiththefactthatbothfortheblackholeandtheusual atbackgroundwithalineardilaton eld,thetimecoordinateshouldbeidenti edwiththetimecoordinateofthematrixmodel.