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

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

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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

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¯ …… 此处隐藏：5740字，全部文档内容请下载后查看。喜欢就下载吧 ……