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

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

hep-th/0408064

Symmetries,ConservedChargesand(Black)Holes

inTwoDimensionalStringTheory

arXiv:hep-th/0408064v1 9 Aug 2004AshokeSenHarish-ChandraResearchInstituteChhatnagRoad,Jhusi,Allahabad211019,INDIAE-mail:ashoke.sen@cern.ch,sen@mri.ernet.inAbstractTwodimensionalstringtheoryisknowntohaveanin nitedimensionalsymmetry,bothinthecontinuumformalismaswellasinthematrixmodelformalism.Wedevelopasystematicprocedureforcomputingtheconservedchargesassociatedwiththesesym-metriesforanycon gurationofD-branesinthecontinuumdescription.WeexpresstheseconservedchargesintermsoftheboundarystateassociatedwiththeD-brane,andalsointermsoftheasymptotic eldcon parisonoftheconservedchargescomputedinthecontinuumdescriptionwiththosecomputedinthematrixmodeldescriptionfacilitatesidenti ingthisweputconstraintsonthecontinuumdescriptionoftheholestatesinthematrixmodel,andmatrixmodeldescriptionoftheblackholessolutionsofthecontin-uumtheory.WealsodiscusspossiblegeneralizationoftheconstructionoftheconservedchargestothecaseofD-branesincriticalstringtheory.

1

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

Contents

1IntroductionandSummary

2SymmetriestoConservedChargesinOpenStringTheory283SymmetriesandConservedChargesinTwoDimensionalStringTheory144ChargesCarriedbytheRollingTachyonBackground

5AsymptoticFieldsProducedbytheRollingTachyon

6RelationtoConservedChargesintheMatrixModel

7CommentsonHoleStates

8µ→0Limit

9ConservedChargesfromAsymptoticStringFieldCon gurations10TwoDimensionalBlackHoles

11LessonsforCriticalStringTheory

LLAPropertiesof|ψ(j),m and|η(j),m 212731333639434850

52 BNormalizationofQj,m

1IntroductionandSummary

Recentinvestigationintwodimensionalstringtheory[1,2,3,4,5]hasshownthattheycanprovideuswithausefularenaforstudyingvariousgeneralpropertiesofstringtheory,mostnotablytherelationshipbetweentheopenandclosedstringdescriptionofunstableD-branesystems[6,7,8,9,10,11].(See[12,13]foraspectsofopen-closedstringdualityforstableD-branesinthisstringtheory.).Thefeaturethatmakesthistheorymostusefulisthatithastwodi erentformulations.The rstone,knownasthecontinuumdescription[14,15](seealso[16]),followstheusualformulationofstringtheorybasedonaworld-sheetactioncontainingamatterpartwithcentralcharge26andaghostpartwithcentralcharge 26.Thematterpartinturnconsistsofafreetime-likescalar eldX0ofcentralcharge1andaLiouvillescalar eld withanexponentiallygrowing

2

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

potential.AlineardilatonbackgroundalongtheLiouvilledirectionmakestheLiouvilletheoryhavetotalcentralcharge25.Inthisformalismthetheorycanbestudiedusingtheusualstringperturbationtheorybasedongenusexpansion.Theotherformulationofthetheory,knownasthematrixmodel,isbasedondiscretizingtheworldsheetpathintegralandtakinganappropriatedoublescalinglimit[17,18,19].Thisinturncanbeshowntobeequivalenttoatheoryoffreenon-interactingfermions,eachmovingunderaninvertedharmonicoscillatorpotential.Thevacuumofthetheoryisastateinwhichalllevelsbelowacertain xedenergyare lledandalllevelabovethisenergyareempty.Inthisformulationwecaneasilyanalyzethesystemtoallordersinperturbationtheory.Theusualclosedstringstatesofthecontinuumstringtheoryarerelatedtothematrixmodelstatesbybosonizationofthefermion eldfollowedbyanon-local eldrede nition[20,21,22].Whileearlyworkonthissubjectfocussedonthecomparisonofthepropertiesofclosedstringsinthetwoformulations,therecentsurgeofinterestinthissubjectarisesfromthestudyofD-branesinthetwodescriptionsofthetheory.ThecontinuumversionofthetheoryadmitsanunstableD0-branecon gurationwithanopenstringtachyononitsworld-volume[23].Followingthegeneralmethoddevelopedin[24,25]onecanconstructanexactclassicalsolutiondescribingtheopenstringtachyonrollingawayfromthemaximumofthepotential.Bystudyingtheclosedstringdescriptionofthisprocessinthecontinuumstringtheoryfollowing[26,27,28],andcomparingthiswiththesinglefermionexcitationinthematrixmodelusingtheknownrelationbetweenthestatesofthematrixmodelandtheclosedstringstatesinthecontinuumdescription,itwasconcludedin[2]thattherollingtachyoncon gurationonasingleD0-braneinthecontinuumtheorydescribespreciselysinglefermionexcitationsinthematrixmodel.Despitethisnewunderstandingoftherelationshipbetweenthematrixmodelandcontinuumdescriptionoftwodimensionalstringtheories,severalquestionsremainunan-swered.Inparticularwestilldonothaveacompletemapbetweentheknownstatesofthecontinuumtheoryandknownstatesofthematrixmodel.Forexamplethematrixmodel,besidescontainingfermionicexcitations,alsocontainsholelikeexcitationswhereweremoveafermionfromanenergylevelbelowthefermilevel.Acompletelyconvinc-ingdescriptionofthesestatesinthecontinuumtheoryisstillmissing(althoughsomecandidateshavebeenproposedin[5,29]).Ontheotherhandthecontinuumversionofthistheoryadmitsblackholesolutions[30,31].AlthoughtherearesomeproposalsforarepresentationoftheEuclideanblackholeinthematrixmodel[32,33],asatisfactorydescriptionoftheseblackholestatesintheLorenzianversionofthematrixmodelisstilllacking.

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