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].