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

TL;DR

This paper unifies the classical and quantum relationships between the KP hierarchy, $\,\hat{W}_{\infty}$ algebra, and the $SL(2,R)/U(1)$ conformal model, introducing a quantum KP hierarchy with commuting charges.

Contribution

It constructs a quantum KP hierarchy by deforming the classical structure via quantization of the $SL(2,R)_k/U(1)$ model and develops quantum $\,\hat{W}_{\infty}$ charges.

Findings

Quantum version of KP hierarchy constructed.

Infinite set of commuting quantum $\,\hat{W}_{\infty}$ charges identified.

Unified classical and quantum description of integrable and conformal models.

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).