Symmetries, Conserved Charges and (Black) Holes in Two Dimen(2)

Boththecontinuumversionandthematrixmodelversionofthetheoryareknowntohavein nitenumberofglobalsymmetries[21,34,35,36,37,38,39,40,41,42,43],and

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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

henceassociatedwiththemtheremustbein nitenumberofconservedcharges.Thusifwecan ndthepreciserelationbetweentheconservedchargesinthecontinuumdescriptionandthoseinthematrixmodeldescription,thencomparisonoftheseconservedchargescouldprovideususefulguidelinesformakingsuitableidenti cationbetweenthestatesinthecontinuumdescriptionandstatesinthematrixmodeldescription.Theconstructionoftheconservedchargesinthematrixmodeldescriptionfollowsstraightforwardapplica-tionofNoether’smethod.Thusthemainissueistoconstructtheseconservedchargesinthecontinuumdescriptionandrelatethemtotheconservedchargesinthematrixmodeldescription.Thisprogramwasinitiatedinapreviouspaper[44]whereitwasshownthatrequirementofBRSTinvarianceconstrainsthetimeevolutionofcertaincomponentsoftheboundarystatedescribingaD-brane,andhenceleadstocertainconservedcharges.Byevaluatingtheseconservedchargesfortherollingtachyoncon guration,andcom-paringthemwiththeconservedchargesofthesamecon gurationinthematrixmodeldescription,wefoundtherelationshipbetweenthesetwosetsofconservedcharges.How-everinthisanalysistheconstructionoftheconservedchargesonthecontinuumsidewassomewhatadhoc,inthesensethatitrequiredassumingacertainstructureoftheboundarystate,andwithinthatstructurerequirementofBRSTinvariance xedthetimedependenceofcertaintermsoftheboundarystate.Ageneralprocedureforconstructingtheconservedchargesforageneralboundarystatewasnotgiven.Alsotheconservedchargesconstructedthiswaydidnotgetrelatedtoanyspeci csymmetryofthetheory.ThemaingoalofthispaperwillbetodevelopasystematicmethodforcomputingthechargecarriedbyaD-braneassociatedwithagivenaglobalsymmetrytransformationinanystringtheory,andthenapplythistotwodimensionalstringtheory.Inparticular,parisonofthechargescarriedbyaD0-braneinthematrixmodelandthecontinuumdescriptionofthetwodimensionalstringtheorythenallowsusto ingtheserelationswecanputconstraintsonwhatkindofD-branedescribestheholestatesofthematrixmodel,andalsowhatkindofmatrixmodelcon gurationdescribestheblackholestatesofthecontinuumstringtheory.

Thepaperisorganizedasfollows.Insection2wereviewhowarigidclosedstringgaugetransformation,forwhichthe eldindependentpartofthetransformationvanishes,generatesglobalsymmetriesoftheopenstring eldtheory[45].Associatedwiththisglobalsymmetrywecanassociateaconservedcharge.WedevelopanalgorithmforcomputingthisconservedchargeforanyD-branesystemintermsoftheboundarystatedescribingtheD-brane.The nalformulafortheconservedchargeisgivenineq.(2.26).Thesimplestglobalsymmetryofthiskindistimetranslation,andtheassociatedconservedchargegives

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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

theenergyoftheD-brane.

Insection3wereviewthestructureofrigidclosedstringgaugetransformationsinthetwodimensionalstringtheory.TheseareobtainedfromtheelementsofrelativeBRSTcohomologyintheghostnumberonesector,andhavebeenclassi edinrefs.[46,36,47,48,42].Therearein nitenumberofsuchrigidgaugetransformations,labelledbySU(2)quantumnumbers(j,m)withj≥1,m= (j 1), (j 2),...(j 1).Thusassociatedwiththesetransformationstherearein nitenumberofconservedchargesQj,ingtheresultofsection2weexplicitlywritedownineqs.(3.31),(3.32)theexpressionforQj,mcarriedbyaD-braneoftwodimensionalstringtheoryintermsoftheboundarystateoftheD-brane.

Insection4weevaluatetheseconservedchargesforaspeci cD-branesystem,–namelytherollingtachyoncon gurationontheunstableD0-braneoftwodimensionalstringtheory.Thesecon gurationsarelabelledbyasingleparameterλ,andwe ndineq.(4.38)explicitexpressionforQj,masafunctionoftheparameterλ.

In[44]wehadcalculatedthediscretestateclosedstringbackgroundproducedbytherollingtachyoncon gurationintheweakcouplingregionoflargenegative .Werecalltheseresultsinsection5(eq.(5.7)),andpointoutthattherearesomeadditionalcontri-butionstothisclosedstringbackgroundduetosomesubtletieswhichwereoverlookedin

[44].Eq.(5.14)givesatypicalexampleofsuchadditionalcontributions.

Thematrixmodeldescriptionofthetheoryalsocontainsanin nitesetofconservedchargesWl,mwith2m∈Z,(l m)∈Z,l≥|m|.Insection6wecomparetheconservedchargesofthematrixmodeldescriptionwiththeconservedchargesofthecontinuumstringtheory.ByevaluatingtheconservedchargesWl,mofsinglefermionexcitationsinthematrixmodel,andcomparingtheλandtimedependenceofthesechargeswiththoseofQj,mcarriedbyasingleD0-braneinthecontinuumdescription,wedetermineineq.(6.6)therelationbetweentheconservedchargesofthematrixmodelandthoseinthecontinuumdescriptionofthetheory.AlthoughtheserelationsarederivedbyevaluatingthechargesinthetwodescriptionsoftheD0-brane,oncetherelationshipisdetermined,itmustholdforanyothersystem.

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