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

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.

29p