Symmetries, Conserved Charges and (Black) Holes in Two Dimen(5)

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)

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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

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].

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