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

ForsimplifyingthenotationweshallregardthevertexoperatorVβasaproductofaleft-chiralvertexoperatorVβLandaright-chiralvertexoperatorVβRofdimension(hβ,0)and(0,hβ)respectively,althoughinthe nalexpressiononlytheproductVβLVβRwillappear.

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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

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

L (where|ηj),m isalinearcombinationofstatescarryingSU(2)quantumnumbers(j 1,m)and(j 2,m),and|τjL 1,m hasSU(2)quantumnumbers(j 1,m).This nishesourdiscussionofchiralBRSTcohomologyinghostnumberszeroand

onesectors.Weshallnowcombinetheleftandtheright-movingstatesmatchingXand momentatoconstructafamilyofstates|Λj,m(p) ofghostnumber1,satisfyingtherequirement(2.21).Wede ne:9LLLL|ψ(j),m ≡|ψ(j),m(p=2m) =mc0|Yj,m +|τj 1,m ,(3.22)

|Λj,m(p) =1

2 LRLR|η(j),m(p) ×|Yj,m(p) +|Oj 1,m(p) ×|ψ(j),m(p)

L |ψ(j),m(p) ×R|Oj 1,m(p) +L|Yj,m(p) ×R|η(j),m(p) .(3.26)

Eqs.(3.24),(3.25)nowgive

(b0 ¯b0)|φj,m(p) =0,¯0)|φj,m(p) =0.(L0 L(3.27)

Forexplicitcomputationoftheconservedchargeinsection4weshallneedtheformof|φj,m(p=2m) .Usingeqs.(3.21),(3.22)weget

LR|φj,m(p=2m) =|Yj,m ×|Yj,m +|ωj,m 1+

2 L|Oj 1,m ×Rc¯0|Yj,m Lc0|Yj,m ×R|Oj 1,m ,(3.28)