Vertex Operators of Sl(2,r) Black Hole and 2-d Gravity

By boosting the vertex operators of Witten's S L(2; R)=U (1) black hole, we show that in the region V they lead to the primary elds of c = 1 matter coupled to gravity at nonzero cosmological constant, while there is no such correspondence in the region I, showing that Witten's black hole corresponds to 2d gravity only in a certain region and in a speciic limit. By using the free eld representation , we will show that the stress tensor of S L(2; R) gauged by its nilpotent subgroup is equivalent to that of the Liouville theory with zero cosmological constant.