On the Kadomtsev–Petviashvili hierarchy, Ŵ∞ algebra, and conformal SL(2,R)/U(1) model. II. The quantum case

Abstract

This article is devoted to constructing a quantum version of the famous Kadomtsev–Petviashvili (KP) hierarchy by deforming its second Hamiltonian structure, namely, the nonlinear Ŵ∞ algebra. This is achieved by quantizing the conformal noncompact SL(2,R)k/U(1) coset model, in which Ŵ∞ appears as a hidden current algebra. For the quantum Ŵ∞ algebra at level k=1, an infinite set of commuting quantum charges in explicit and closed form was successfully constructed. Using them, a completely integrable quantum KP hierarchy is constructed in the Hamiltonian form. A two-boson realization of the quantum Ŵ∞ currents has played a crucial role in this exploration.