On the KP Hierarchy, ${\hat{W}}_{\infty }$ Algebra, and Conformal SL(2,R)/U(1) Model: II. The Quantum Case

Abstract

This paper is devoted to constructing a quantum version of the famous KP hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear ${\hat{W}}_{\infty }$ algebra. This is achieved by quantizing the conformal noncompact $SL(2,R{)}_k/U(1)$ coset model, in which ${\hat{W}}_{\infty }$ appears as a hidden current algebra. For the quantum ${\hat{W}}_{\infty }$ algebra at level $k=1$, we have succeeded in constructing an infinite set of commuting quantum charges in explicit and closed form. Using them a completely integrable quantum KP hierarchy is constructed in the Hamiltonian form. A two boson realization of the quantum ${\hat{W}}_{\infty }$ currents has played a crucial role in this exploration.

33p