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

TL;DR

This paper explores the deep connections between the KP hierarchy, $ abla ext{W}_{ ext{infinity}}$ algebra, and the $SL(2,R)/U(1)$ conformal model at the classical level, revealing hidden symmetries and integrable structures.

Contribution

It explicitly derives the Poisson brackets of the KP hierarchy's Hamiltonian structure and realizes the $ ext{W}_{ ext{infinity}}$ algebra within the $SL(2,R)/U(1)$ model using free bosons.

Findings

$ ext{W}_{ ext{infinity}}$ algebra is realized as a hidden current algebra in the coset model.

Infinite KP flows preserve the $ ext{W}_{ ext{infinity}}$ algebra in the model.

Explicit free field realization of higher-spin currents in the coset model.

Abstract

In this paper we study the inter-relationship between the integrable KP hierarchy, nonlinear $\hat{W}_{\infty}$ algebra and conformal noncompact $SL(2,R)/U(1)$ coset model at the classical level. We first derive explicitly the Possion brackets of the second Hamiltonian structure of the KP hierarchy, then use it to define the $\hat{W}_{1+\infty}$ algebra and its reduction $\hat{W}_{\infty}$. Then we show that the latter is realized in the $SL(2,R)/U(1)$ coset model as a hidden current algebra, through a free field realization of $\hat{W}_{\infty}$, in closed form for all higher-spin currents, in terms of two bosons. An immediate consequence is the existence of an infinite number of KP flows in the coset model, which preserve the $\hat{W}_{\infty}$ current algebra.