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

TL;DR

This paper constructs a quantum version of the KP hierarchy by deforming the $ olinebreak ext{W}_{ olinebreak ext{infinity}}$ algebra through quantization of the $SL(2,R)/U(1)$ coset model, leading to an integrable quantum hierarchy.

Contribution

It introduces a quantized $ ext{W}_{ olinebreak ext{infinity}}$ algebra and constructs a corresponding integrable quantum KP hierarchy with explicit quantum charges.

Findings

Successfully constructed quantum $ ext{W}_{ olinebreak ext{infinity}}$ algebra at level 1.

Derived an explicit set of commuting quantum charges.

Established a quantum integrable KP hierarchy.

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.