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

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.

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 eldt …… 此处隐藏：6578字，全部文档内容请下载后查看。喜欢就下载吧 ……