On the KP Hierarchy, ${\hat{W}}_{\infty }$ Algebra, and Conformal SL(2,R)/U(1) Model --- The Classical and Quantum Cases

Abstract

We give a unified description of our recent results on the the inter-relationship between the integrable infinite KP hierarchy, nonlinear ${\hat{W}}_{\infty }$ current algebra and conformal noncompact $SL(2,R)/U(1)$ coset model both at the classical and quantum levels. In particular, we present the construction of a quantum version of the KP hierarchy by deforming the second KP Hamiltonian structure through quantizing the $SL(2,R{)}_k/U(1)$ model and constructing an infinite set of commuting quantum ${\hat{W}}_{\infty }$ charges (at least at $k$=1).

(Invited talk given the second at the XXI International Conference on Differential Geometric Methods in Theoretical Physics, Nankai Institute of Mathematics, Tianjin, China; June 5-9, 1992; to appear in Proceedings.), 12p, Latex file