Symmetries, Conserved Charges and (Black) Holes in Two Dimen(8)

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)

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