Symmetries, Conserved Charges and (Black) Holes in Two Dimen(4)

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

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