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.