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.

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

3

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

4

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

5

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

6

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.

Althoughthemainemphasisofthepaperhasbeenontheconstructionofthecon-servedchargesandtheirinterpretationinthetwodimensionalstringtheory,wecantrytogeneralizetheconstructiontocriticalstringtheorybyreplacingtheprimaryvertexoper-atorsintheLiouville eldtheorybyappropriateprimaryvertexoperatorsinthecriticalstringtheory,andbyreplacingtheLiouvilleVirasorogeneratorsbythetotalVirasorogeneratorsassociatedwithallthe25space-likecoordinate eldsinthecriticalstringthe-ory.Therearehowevervarioussubtleissuesinthisapproach.Therearediscussedinsection11.

Finallytheappendicescontainsometechnicalresultswhicharerequiredfortheex-plicitconstructionofconservedchargesandtheirnormalizationintwodimensionalstringtheory.

7

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

2SymmetriestoConservedChargesinOpenString

Theory

Inthissectionwebrie youtlinethegeneralprocedureforobtainingtheconservedchargeinclassicalopenstringtheoryassociatedwithaspeci cglobalsymmetry.Weshallfocusontheglobalsymmetriesassociatedwithrigidgaugetransformationsinclosedstringtheory[45].Anexampleofthisisspace-timetranslationsymmetry,whichcanbethoughtofasarigidgeneralcoordinatetransformation.

Weshallcarryoutthediscussioninthecontextofstring eldtheory.Webeginwithsomeversionofcovariantopen-closedstring eldtheory[45]formulatedforagivenD-braneinagivenspace-timebackground.Howeverouranalysiswillbequitegeneralandweshallnotrestrictourselvestoanyspeci cformoftheaction.Letusdenoteby{Φα}theclosedstringdegreesoffreedomandby{Ψr}theopenstringdegreesoffreedom,withtheindicesαandr,besidescontainingdiscretelabels,alsocarryinginformationaboutmomentaofthe eldsalongnon-compactspace-timedirections.Thentheopen-closedstring eldtheoryactionhastheform[45]:

1

gs0S1(Φ,Ψ)+O(gs),(2.1)

2 1wheregsdenotesstringcouplingconstant.Theordergsandgstermsgetcontributions

respectivelyfromthesphereanddiskcorrelationfunctionsoftheworld-sheettheory.LetDdenotethedimensionofspace-timeinwhichtheclosedstringtheorylives.Thenatypicalclosedstringgaugetransformationisparametrizedbysomearbitraryfunction (p)ofDdimensionalmomentump.Thein nitesimalgaugetransformationlawstaketheform:

δΦα=

δΨr=n=0∞ ∞ ngs

ngsn)δΦ(αn)δΨ(r== dp (p)dp (p)DD

n=0 h(0)α(Φ,p)+gsh(1)α(Φ,Ψ,p)+O(gs), +2O(gs) (0)fr(Φ,Ψ,p)(2.2)

n)(n)(0)forsuitablefunctionh(αandfr.Thecontributionstohαcomefromspherecorrelation

(0)functions,whereasthecontributionstoh(1)comefromdiskcorrelationfunctions.αandfr

NotethattheleadingcontributiontotheactionandtheleadingcontributiontoδΦαdonotdependontheopenstring eldsΨr.Wecallthisactionandgaugetransformationlawstreelevelclosedstringactionandgaugetransformationlawsrespectively.Ontheotherhand1Sopen(Ψ)≡

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

iscalledthetreelevelopenstring eldtheoryactionintheΦ=0closedstringbackground.1Invarianceofthefullaction(2.1)underthegaugetransformationlaws(2.2)gives:

δS0(Φ)

δS1(Φ,Ψ)(2.5)δΦαδΦα

etc.Weshallchoosethestring eldvariablessuchthatΦ=0istriviallyasolutionoftheclassicalclosedstring eldequations.ThusδS0/δΦαvanishesatΦ=0.PuttingΦ=0ineq.(2.5)andusing(2.2),(2.3)weget

δΦ(0)α+δΦ(1)α=0δS1(Φ,Ψ)

δΨr(0)fr(Φ=0,Ψ,p)=0. (2.6)

Ingeneralh(0)α(Φ=0,p)isnon-zero.Howeversupposeforsomespecialvalueofthemomentumpitvanishes:

h(0)(2.7)α(Φ=0,p=c)=0.

Physicallyitmeansthatthe eldindependentterminthetreelevelclosedstringgaugetransformationlawvanishes.InotherwordswehavearigidgaugetransformationthatleavestheΦ=0backgroundunchanged.Puttingp=cin(2.6)wenowget

δSopen(Ψ)

Afamilyofopenclosedstring eldtheorywasconstructedin[45],andWitten’sopenstring eldtheory[50]appearsastheopenstringsectorofaspecialmemberofthisfamily.1

9

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

Howeverthisprocedureonlygivesthedi erencebetweentheconservedchargecarriedbyagivenopenstring eldcon guration,andthatcarriedbytheΨ=0con gurationrepresentingtheoriginalD-braneonwhichwehaveformulatedtheopenstring eldtheory.Inparticularifwesettheopenstring eldΨtozero,thentheexpressionfortheconservedchargevanishes.OurmaininterestontheotherhandwillbeintheexpressionfortheconservedchargethattheoriginalD-branecarries.Forthisweneedtouseadi erentmethodwhichweshalldescribenow.

ThebasicprocedurecanbeunderstoodinanalogywiththecomputationoftheenergymomentumtensorofaD-brane.De ningtheenergy-momentumtensorintheopenstring eldtheorythroughtheNoetherprescriptiongivescorrectlythedi erenceintheenergy-momentumtensorbetweentwoopenstringcon gurations[51],butthisdoesnotgivetheenergy-momentumtensoroftheD-braneitself.ThelattercanbecalculatedbyexaminingthecouplingofthemetrictotheD-braneworld-volume.InasimilarspiritonewouldexpectthattheinformationaboutallotherconservedchargescarriedbytheD-brane,whichareassociatedwithrigidgaugetransformationsthatleavetheclosedstringbackgroundΦ=0invariant,shouldalsobecalculablebyexaminingthecouplingofthevariousclosedstringmodestotheD-brane.Infacttherelevantinformationisalreadycontainedineq.(2.6).Usingeq.(2.7)andassumingthath(0)α(Φ=0,p)isanalyticatp=c,wecanwrite

µh(0)(2.10)α(Φ=0,p)=(pµ cµ)hα(p),

µ.Ifwede neforsomehα

Gα(Ψ)=δS1(Φ,Ψ)

δΨr(0)fr(Φ=0,Ψ,p)=0.(2.12)

Nowifthe eldsΨssatisfytheirequationsofmotionthenδSopen(Ψ)/δΨr=0.Inthiscasewehave

µ(p)=0.(pµ cµ)Gα(Ψ)h(2.13)α

Ifwede ne

F(x)=

then(2.13)mayberewrittenasµ µ(p)dDpe ip.xGα(Ψ)hα

ic.x(2.14) µe F(x)=0.µ (2.15)

10

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

Thisgivestheconservedcharge

dD 1xeic.xF0(x).(2.16)

Weshallnowmakethisconstructionmoreexplicitbyworkingwithspeci crepresen-tationofclosedstring elds.Weshallrestrictouranalysistoaclosedstringbackgroundinwhichthetimedirectionhasassociatedwithitaworld-sheetconformal eldtheory(CFT)ofafreescalar eldX0whichdoesnotcoupletoanyotherworld-sheet eld.InthiscasewecanworkwiththeEuclideancontinuationofthetheoryobtainedbythereplacementx0→ ix.Wecanrepresenttheclosedstring eldbyastate|Φ ofghostnumbertwointhebulkCFTonacylinder,satisfying

(b0 ¯b0)|Φ =0,¯0)|Φ =0,(L0 L(2.17)

¯ndenotethetotalwherebn,¯bn,cn,c¯ndenotetheusualghostoscillators,andLn,L

Virasorogeneratorsoftheworld-sheettheoryofmatterandghost elds.Closedstringgaugetransformationsinthistheoryaregeneratedbyghostnumberonestates|Λ oftheCFTonacylinder,satisfying

(b0 ¯b0)|Λ =0,¯0)|Λ =0.(L0 L(2.18)

Thee ectofthein nitesimalgaugetransformationsontheclosedstring eldsisgivenby:¯B)|Λ +O(Φ),δ|Φ =(QB+Q(2.19)

¯BaretheholomorphicandantiholomorphiccomponentsoftheBRSTwhereQBandQ

chargeinclosedstringtheory.Thusfora|Λ satisfying

¯B)|Λ =0,(QB+Q(2.20)

thein nitesimalgaugetransformationof|Φ vanishesat|Φ =0.Asaresultthisgaugetransformationleavesthe|Φ =0backgroundunchanged.Byourpreviousargument,thismustgenerateasymmetryofthepureopenstring eldtheorylivingonaD-brane,andgiverisetoaconservedchargeinthistheory.2

OurgoalwillbetoconstructexpressionsfortheseconservedchargesexplicitlyforanyD-branesystemlivinginthisclosedstringbackground.Forsimplicityweshallevaluate

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

thechargeintrivialopenstringbackgroundΨ=0,–thiswillevaluatethechargecarriedbythespeci cD-braneusedintheconstructionoftheopenclosedstring eldtheorywithoutanyfurtheropenstringexcitationsonthebrane.Let|Λ(p) denoteafamilyofclosedstringgaugetransformationparameterslabelledbyXmomentump,suchthat¯B)|Λ(p) vanishesatp=c|Λ(p=c) =|Λ forsomespecialmomentumc.3Then(QB+Q

andwecanwrite¯B)|Λ(p) =(p c)|φ(p) ,(QB+Q(2.21)

¯B)isnilpotent,wewhere|φ(p) issomeghostnumbertwostate.Nowsince(QB+Q

seefromeq.(2.21)that|φ(p) isBRSTinvariantforallp=c,andhencebyanalyticcontinuationBRSTinvariantalsoforp=c.Furthermoreithasthepropertythatforanyp=citisBRSTtrivial,butforp=citcouldbeanon-trivialelementoftheBRSTcohomologyintheghostnumbertwosector.Weshallseelaterthatwecangetnon-trivialconservedchargesonlyif|φ(p=c) isnotBRSTtrivial.

Letusdenoteby|B theboundarystateassociatedwiththeD-braneonwhichwehaveformulatedtheopenstringtheory.Thenthefullstring eldtheoryactioncontainsacoupling:

B|(c0 c¯0)|Φ .(2.22)

Invarianceofthistermunderthein nitesimalgaugetransformation(2.19)generatedbythefamilyofgaugetransformationparameters|Λ(p) requires:

¯B)|Λ(p) =0, B|(c0 c¯0)(QB+Q(2.23)

ForordinaryD-braneseq.(2.23)followsfromtheBRSTinvarianceof B|andtheanalogsof(2.17),(2.18):

¯B)=0, B|(QB+Q B|(b0 ¯b0)=0,¯0)=0. B|(L0 L(2.24)

¯B)by{(c0 c¯B)}in(2.23).ThisThisallowsustoreplace(c0 c¯0)(QB+Q¯0),(QB+Q

doesnothaveanyzeromodeof(c0 c¯0)ing(2.21),eq.(2.23)becomes:

(p c) B|(c0 c¯0)|φ(p) =0.

Ifwede ne:

F(x)=

3(2.25)Weareassumingthattheothermomentumcomponentshavealreadybeensetequaltothespeci c¯B)|Λ(p) vanishes.valuesforwhich(QB+Q dp

12

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

then(2.25)mayberewrittenas

xeF(x)=0.

Replacingxbyix0wenowget:

icx (2.27) 0e

0 cx0F(ix)=0.0 (2.28)Thuse cxF(ix0)isaconservedcharge.Thisgivesageneralprocedureforconstructing

theconservedchargecarriedbyaD-branecorrespondingtoaspeci crigidgaugetrans-formationinclosedstringtheory.ThesuggestionthattheBRSTinvarianceof B|carriesinformationaboutconservedchargeshasbeenmadeearlierin[52].

Givenanelement|Λ oftheBRSTcohomologywithaspeci cmomentumc,thereareclearlyin nitenumberoffamilies|Λ(p) withthepropertythat|Λ(p=c) =|Λ .Onphysicalgroundstheconservedchargeassociatedwiththesymmetrygeneratedby|Λ shouldnotdependonthechoiceofthefamily.Weshallnowprovethisexplicitlybydemonstratingthatiftwofamiliesofgaugetransformationsparameters|Λ(1)(p) and|Λ(2)(p) approachthesamevalueatp=c,thentheygiverisetothesameconservedcharge.Inthiscase,wemaywrite

|Λ(1)(p) |Λ(2)(p) =(p c)|Λ(0)(p) ,(2.29)

forsome|Λ(0)(p) ,sothatthedi erencebetween|Λ(1)(p) and|Λ(2)(p) vanishesatp=c.Eqs.(2.21)and(2.29)nowgive:

¯B)|Λ(0)(p) ,|φ(1)(p) |φ(2)(p) =(QB+Q(2.30)

where|φ(i)(p) isrelatedto|Λ(i)(p) asineq.(2.21).IfF(1)(x)andF(2)(x)denotethecorrespondingconservedchargesasde nedin(2.26),thenwehave

F(1)(x) F(2)(x)= dp

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

someghostnumberzerostate|χ .Let|χ(p) denoteafamilyofstateslabelledbythemo-

p) ≡(Q+Q¯B)|χ(p) hasthementumpsuchthat|χ(p=c) =|χ .Thenthefamily|Λ(B

propertythatitreducesto|Λ forp=c.Thuswecancomputetheconservedchargeasso-

p) .Howeverinthiscase(Q+Q p) ¯B)|Λ(ciatedwiththissymmetryusingthisfamily|Λ(B (p) =(p m) 1(Q+Q p) ¯B)|Λ(vanishesforallp,andhencethecorrespondingstate|φB

ingthede nition(2.26)oftheconservedchargeweseeclearlythatthecorrespondingconservedchargealsovanishesinthiscase.

Finallywenotefromthede nition(2.26)ofF(x)andthefactthatF(x)∝e icxduetotheconservationlaw,thatthevalueofFdependsonthematrixelement B|(c0 c¯0)|φ(p) atp=c.If|φ(p=c) isBRSTexactthenthismatrixelementvanishesandwedonotgetanon-trivialconservedcharge.

3SymmetriesandConservedChargesinTwoDi-

mensionalStringTheory

Inthissectionweshallusetheresultsofsection2toconstructin nitenumberofconservedchargesintwodimensionalbosonicstringtheory.Webeginwithabriefreviewoftwodimensionalstringtheory.Theworld-sheetdescriptionofthetheoryinvolvesatimelikescalar eldX0,aLiouville eldtheorywithc=25andtheusualghost eldsb,c,¯b,c¯.Intheα′=1unitthatweshallbeusing,theLiouvilletheoryisdescribedbyasinglescalar eld withexponentialpotentialintheworldsheetaction:4

sliouville= dz2 1

TheLiouviletheorywithc=25actuallyhasaterm∝ e2 intheworld-sheetaction[53,54].Asin[1,2,3]weshallregardthec=25Liouvilletheoryasthec→25limitoftheorieswithc>25.Forc>25,(3.1)(withe2 replacedbyanappropriatepowerofe )isthecorrectformoftheaction,butµundergoesanin niterenormalizationaswetakethec→25limit.4

14

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

oftheworld-sheetscalar eldX=iX0andtheLiouville eld asindependent,con-structstatesintheleft-andtheright-movingsectorsseparately,andthencombinethemmatchingthemomentaintheleft-andtheright-movingsectortoconstructproperstatesofthetwodimensionalstringtheory.WebeginwiththeCFTassociatedwiththefreescalar eldX.LetusdenotebyXLandXRtheleftandtheright-movingcomponentsofX.TheCFT,besidescontainingtheusualprimarystateseikXL(0)|0 XandeikXR(0)|0 X,containsasetofprimaries|j,m L,|j,m Roftheform[55]:

L|j,m L=Pj,me2imXL(0)|0 X,R2imXR(0)|j,m R=Pj,me|0 X,(3.3)

LRwherePj,mandPj,maresomecombinationofnon-zeromodeXL,XRoscillatorsoflevel(j2 m2),and(j,m)areSU(2)quantumnumberswith j≤m≤j.5Forexample,wehave|1,0 L=α 1|0 X,|1,0 R=α¯ 1|0 X,whereαn,α¯naretheusualoscillatorsof

LRtheX- eld.Pj,±jandPj,±j,beingoflevel0,mustbeidentityoperators.Thus|j,j L=

e2ijXL(0)|0 ,|j,j R=e2ijXR(0)|0 .Weshallcombinetheleftandtheright-movingmodestode ne:

LR|j,m X=|j,m L×|j,m R=Pj,mPj,me2imX(0)|0 X.(3.4)

Infactthistheorycontainsamoregeneralsetofprimaries|j,m L×|j′,m R,butweshallnotintroduceaspecialsymboltolabelthesestates.Forlateruseweshallalsode ne:

L|j,m,p L=Pj,meipXL(0)|0 X,R|j,m,p R=Pj,meipXR(0)|0 X,(3.5)

foranarbitraryX-momentump,and

LR|j,m,p X=|j,m,p L×|j,m,p R=Pj,mPj,m|p X,(3.6)

(3.7)where

LRWeshallnormalizePj,m,Pj,msuchthat:

X j,m,p|j′|p X=eipX(0)|0 X.′X p|p X,m,p′ X=δjj′=2πδ(p+p′)δjj′,(3.8)

whereX ·|· XdenotesBPZinnerproductintheCFToftheX- eld.Thevanishingofthisinnerproductforj=j′followssimplyfromthefactthatthetwostateshavedi erentconformalweights.

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

TheLiouville eldtheorycontainsasetofprimaryvertexoperatorsVβofconformalweight(hβ,hβ)with1hβ=

Γ( iP)

Γ( iP) 2Thesecondterminthisexpressionisrequiredbythere ectionsymmetry[58,59]V2+iP≡ 2δ(P P′) . (3.10)

Γ( iP)e(2 iP) ,(3.11)

forlargenegative .Thesecondtermre ectsthee ectoftheexponentiallygrowingpotentialforlargepositive .

Thisnormalizationdi ersfromtheimplicitnormalizationassumedin[44]wheretheseconddeltafunctionin(3.10)wasabsent,andwepretendedthatV2+iPandV2 iPareindependentvertexoperators.AslongasweworkwithappropriatelinearcombinationsofV2+iPandV2 iPwhichobeythere ectionsymmetry,thisproceduregivesthecorrectresult.TheclosestanalogyofthisinordinaryquantummechanicsisthatwhilestudyingafreeparticleonahalflinewithNeumann(Dirichlet)boundaryconditiononthewave-functionattheorigin,wecanstudythetheoryonthefulllinewithbasisstateseikxandattheendrestrictthe eldcon gurationstobeeven(odd)underre ectionaroundtheorigin.IncontrastinthispaperweusetheconventionthatV2+iPitselfobeysthere ectionsymmetrydictatedbytheCFT.Thisisanalogoustousing2cosx(2sinx)asbasisfunctionsforfreeparticleonahalflinewithNeumann(Dirichlet)boundaryconditiononthewave-functionattheorigin.Anybasisindependentrelatione.g.eq.(5.7)willnotbea ectedbythisdi erenceinthechoiceofbasis.

ForsimplifyingthenotationweshallregardthevertexoperatorVβasaproductofaleft-chiralvertexoperatorVβLandaright-chiralvertexoperatorVβRofdimension(hβ,0)and(0,hβ)respectively,althoughinthe nalexpressiononlytheproductVβLVβRwillappear.

16

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

WeshallnowreviewsomeresultsonthechiralBRSTcohomologyoftheworld-sheettheory[46,47,48,36,42,5,61,62]inghostnumberzeroandonesectors.6Asweshallsee,thesewillbethebasicbuildingblocksfortheconstructionofnon-trivialsymmetrygeneratorsofthetwodimensionalstringtheoryunderwhichD-branesarecharged.Forde nitenessweshalldescribetheresultsintheleft-moving(holomorphic)sector,butidenticalresultsholdintheright-movingsectoraswell.Webeginintheghostnumberonesector.Inthissectorwehaveanin nitenumberofelementsoftheBRSTcohomologylabelledbytheSU(2)quantumnumbers(j,m)with j≤m≤j,representedbythestates

LL|Yj,m =|j,m L V2(1 j)(0)|0 liouville c1|0 ghost

LL=Pj,me2imXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost,(3.12)

LwherePj,mhasbeende nedin(3.3).ByconstructionthesestateshavezeroL0eigenvalue.Forlaterusewede ne:

LLL|Yj,m(p) =Pj,meipXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost.(3.13)

Intheghostnumber0sectoralsowehaveanin nitenumberofelementsoftheBRSTcohomologylabelledbytheSU(2)quantumnumbers(j 1,m)with (j 1)≤m≤j 1.TherepresentativeelementsoftheBRSTcohomologycanbechosentobeoftheform

LLL|Oj 1,m =Qj 1,m|j 1,m L V2(1 j)(0)|0 liouville c1|0 ghost,(3.14)

whereQLj 1,misanoperatorofghostnumber 1,level(2j 1)constructedfromnegativemodedghostoscillatorsandXandLiouvilleVirasorogenerators[61].Using(3.3)thiscanberewrittenas

LL2imXL(0)L|Oj|0 X V2(1 1,m =Rj 1,me j)(0)|0 liouville c1|0 ghost(3.15)

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

LLwhereRLj 1,m≡Qj 1,mPj 1,misanoperatorofghostnumber 1constructedfrom

negativemodedXandghostoscillatorsandLiouvilleVirasorogenerators.7Forlaterusewenowde ne:

LLipXL(0)L|Oj|0 X V2(1 1,m(p) =Rj 1,me j)(0)|0 liouville c1|0 ghost.(3.16)

LLNotethat|Yj,m(p) and|Oj 1,m(p) arebothbuiltbytheactionofXandghostoscillators

LandLiouvilleVirasorogeneratorsonthesameFockvacuumeipXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost,andsatisfy

Lb0|Oj 1,m(p) =0,Lb0|Yj,m(p) =0.(3.17)

LLFurthermore,since|Yj,m(p=2m) and|Oj 1,m(p=2m) havezeroL0eigenvalues,we

have

LL0|Oj 1,m(p) =1

4(3.18)

LLLLGiventhat|Oj 1,m =|Oj 1,m(p=2m) and|Yj,m =|Yj,m(p=2m) areBRST

invariant,wemusthave8

LLQB|Oj 1,m(p) =(p 2m)|η(j),m(p) ,L(p2 4m2)|Yj,m(p) .(3.19)

(3.20)and

LLforsomestates|η(j),m(p) and|ψ(j),m(p) .Itfollowsfromeqs.(3.19)and(3.20)andthe

LLnilpotenceofQBthatboth|η(j),m(p) and|ψ(j),m(p) areBRSTinvariantforanyp=2m

andhencebyanalyticcontinuationalsoforp=2m.Wealsoseefromeqs.(3.19),(3.20)

LLthatforp=2m,|η(j),m(p) and|ψ(j),m(p) areBRSTtrivialbutforp=2mtheycanbe

LLBRSTnon-trivial.Finallywenotethatsince|Oj 1,m(p) and|Yj,m(p) havenon-vanishing

LLL0eigenvaluesproportionalto(p2 4m2),|η(j),m(p) and|ψ(j),m(p) de nedthrough(3.19)

and(3.20)arenotannihilatedbyb0ingeneral.

IthasbeenshowninappendixAthatatp=2m,

LLLL (|η(j),m ,j),m ≡|η(j),m(p=2m) =|Yj,m +|ηLLQB|Yj,m(p) =(p 2m)|ψ(j),m(p) ,(3.21)

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

L (where|ηj),m isalinearcombinationofstatescarryingSU(2)quantumnumbers(j 1,m)and(j 2,m),and|τjL 1,m hasSU(2)quantumnumbers(j 1,m).This nishesourdiscussionofchiralBRSTcohomologyinghostnumberszeroand

onesectors.Weshallnowcombinetheleftandtheright-movingstatesmatchingXand momentatoconstructafamilyofstates|Λj,m(p) ofghostnumber1,satisfyingtherequirement(2.21).Wede ne:9LLLL|ψ(j),m ≡|ψ(j),m(p=2m) =mc0|Yj,m +|τj 1,m ,(3.22)

|Λj,m(p) =1

2 LRLR|η(j),m(p) ×|Yj,m(p) +|Oj 1,m(p) ×|ψ(j),m(p)

L |ψ(j),m(p) ×R|Oj 1,m(p) +L|Yj,m(p) ×R|η(j),m(p) .(3.26)

Eqs.(3.24),(3.25)nowgive

(b0 ¯b0)|φj,m(p) =0,¯0)|φj,m(p) =0.(L0 L(3.27)

Forexplicitcomputationoftheconservedchargeinsection4weshallneedtheformof|φj,m(p=2m) .Usingeqs.(3.21),(3.22)weget

LR|φj,m(p=2m) =|Yj,m ×|Yj,m +|ωj,m 1+

2 L|Oj 1,m ×Rc¯0|Yj,m Lc0|Yj,m ×R|Oj 1,m ,(3.28)

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