Γ( iP)
Γ( iP) 2Thesecondterminthisexpressionisrequiredbythere ectionsymmetry[58,59]V2+iP≡ 2δ(P P′) . (3.10)
Γ( iP)e(2 iP) ,(3.11)
forlargenegative .Thesecondtermre ectsthee ectoftheexponentiallygrowingpotentialforlargepositive .
Thisnormalizationdi ersfromtheimplicitnormalizationassumedin[44]wheretheseconddeltafunctionin(3.10)wasabsent,andwepretendedthatV2+iPandV2 iPareindependentvertexoperators.AslongasweworkwithappropriatelinearcombinationsofV2+iPandV2 iPwhichobeythere ectionsymmetry,thisproceduregivesthecorrectresult.TheclosestanalogyofthisinordinaryquantummechanicsisthatwhilestudyingafreeparticleonahalflinewithNeumann(Dirichlet)boundaryconditiononthewave-functionattheorigin,wecanstudythetheoryonthefulllinewithbasisstateseikxandattheendrestrictthe eldcon gurationstobeeven(odd)underre ectionaroundtheorigin.IncontrastinthispaperweusetheconventionthatV2+iPitselfobeysthere ectionsymmetrydictatedbytheCFT.Thisisanalogoustousing2cosx(2sinx)asbasisfunctionsforfreeparticleonahalflinewithNeumann(Dirichlet)boundaryconditiononthewave-functionattheorigin.Anybasisindependentrelatione.g.eq.(5.7)willnotbea ectedbythisdi erenceinthechoiceofbasis.
ForsimplifyingthenotationweshallregardthevertexoperatorVβasaproductofaleft-chiralvertexoperatorVβLandaright-chiralvertexoperatorVβRofdimension(hβ,0)and(0,hβ)respectively,althoughinthe nalexpressiononlytheproductVβLVβRwillappear.
16
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
WeshallnowreviewsomeresultsonthechiralBRSTcohomologyoftheworld-sheettheory[46,47,48,36,42,5,61,62]inghostnumberzeroandonesectors.6Asweshallsee,thesewillbethebasicbuildingblocksfortheconstructionofnon-trivialsymmetrygeneratorsofthetwodimensionalstringtheoryunderwhichD-branesarecharged.Forde nitenessweshalldescribetheresultsintheleft-moving(holomorphic)sector,butidenticalresultsholdintheright-movingsectoraswell.Webeginintheghostnumberonesector.Inthissectorwehaveanin nitenumberofelementsoftheBRSTcohomologylabelledbytheSU(2)quantumnumbers(j,m)with j≤m≤j,representedbythestates
LL|Yj,m =|j,m L V2(1 j)(0)|0 liouville c1|0 ghost
LL=Pj,me2imXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost,(3.12)
LwherePj,mhasbeende nedin(3.3).ByconstructionthesestateshavezeroL0eigenvalue.Forlaterusewede ne:
LLL|Yj,m(p) =Pj,meipXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost.(3.13)
Intheghostnumber0sectoralsowehaveanin nitenumberofelementsoftheBRSTcohomologylabelledbytheSU(2)quantumnumbers(j 1,m)with (j 1)≤m≤j 1.TherepresentativeelementsoftheBRSTcohomologycanbechosentobeoftheform
LLL|Oj 1,m =Qj 1,m|j 1,m L V2(1 j)(0)|0 liouville c1|0 ghost,(3.14)
whereQLj 1,misanoperatorofghostnumber 1,level(2j 1)constructedfromnegativemodedghostoscillatorsandXandLiouvilleVirasorogenerators[61].Using(3.3)thiscanberewrittenas
LL2imXL(0)L|Oj|0 X V2(1 1,m =Rj 1,me j)(0)|0 liouville c1|0 ghost(3.15)
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
LLwhereRLj 1,m≡Qj 1,mPj 1,misanoperatorofghostnumber 1constructedfrom
negativemodedXandghostoscillatorsandLiouvilleVirasorogenerators.7Forlaterusewenowde ne:
LLipXL(0)L|Oj|0 X V2(1 1,m(p) =Rj 1,me j)(0)|0 liouville c1|0 ghost.(3.16)
LLNotethat|Yj,m(p) and|Oj 1,m(p) arebothbuiltbytheactionofXandghostoscillators
LandLiouvilleVirasorogeneratorsonthesameFockvacuumeipXL(0)|0 X V2(1 j)(0)|0 liouville c1|0 ghost,andsatisfy
Lb0|Oj 1,m(p) =0,Lb0|Yj,m(p) =0.(3.17)
LLFurthermore,since|Yj,m(p=2m) and|Oj 1,m(p=2m) havezeroL0eigenvalues,we
have
LL0|Oj 1,m(p) =1
4(3.18)
LLLLGiventhat|Oj 1,m =|Oj 1,m(p=2m) and|Yj,m =|Yj,m(p=2m) areBRST
invariant,wemusthave8
LLQB|Oj 1,m(p) =(p 2m)|η(j),m(p) ,L(p2 4m2)|Yj,m(p) .(3.19)
(3.20)and
LLforsomestates|η(j),m(p) and|ψ(j),m(p) .Itfollowsfromeqs.(3.19)and(3.20)andthe
LLnilpotenceofQBthatboth|η(j),m(p) and|ψ(j),m(p) areBRSTinvariantforanyp=2m
andhencebyanalyticcontinuationalsoforp=2m.Wealsoseefromeqs.(3.19),(3.20)
LLthatforp=2m,|η(j),m(p) and|ψ(j),m(p) areBRSTtrivialbutforp=2mtheycanbe
LLBRSTnon-trivial.Finallywenotethatsince|Oj 1,m(p) and|Yj,m(p) havenon-vanishing
LLL0eigenvaluesproportionalto(p2 4m2),|η(j),m(p) and|ψ(j),m(p) de nedthrough(3.19)
and(3.20)arenotannihilatedbyb0ingeneral.
IthasbeenshowninappendixAthatatp=2m,
LLLL (|η(j),m ,j),m ≡|η(j),m(p=2m) =|Yj,m +|ηLLQB|Yj,m(p) =(p 2m)|ψ(j),m(p) ,(3.21)
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