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=