CONFORMAL INVARIANCE AND TWO DIMENSIONAL QUANTUM GRAVITY

Wafic  A. SABRA

Working within the framework of theories with induced gravity developed by Knizhnik, Polyakov and Zamolodchikov, we consider the general effects of perturbation by operators constructed from primary matter fields and which break scale invariance. We find constraints on possible renormalization group flows induced by this perturbation. We also extend our results to the theory of induced two dimensional (1,0) Supergravity where we consider the effects of perturbing the theory with relevant operators that preserve rotational, translational and supersymmetry whilst breaking scale invariance. In particular we calculate the correlation functions {J^a(x)J^b(y)} of graded SL(2,R) currents J^a (x) in the presence of such perturbations from which we define central charge functions K. These functions are shown to be monotonic increasing and are the analogue of Zamolodchikov's C-function as defined in conformal field theories.

We also consider the induced (2, 0)Supergravity theory in the superchiral gauge and derive the corresponding OSp(2,2) current algebra. The renormalization of the central charge of the current algebra and the physical primary state conditions of the fields are derived using the residual gauge transformations and BRST quantization .

Finally , we examine the relation between the Borel gauged OSp(N,2) Wess-Zumino-Novikov-Witten models and Polyakov's two dimensional quantum supergravities. We show the equivalence between these models and the symplectic actions derived on the coadjoint orbits of the N-extended super Virasoro groups.