W-infinity structure of the $sl(N)$ conformal affine Toda theories

TL;DR

This paper explores the $W_{ty}$ symmetry in $sl(N)$ Conformal Affine Toda theories, clarifying their relation to the KP hierarchy and discussing potential links to matrix models.

Contribution

It demonstrates how to reduce the zero curvature equation to a Lax equation with $W_{ty}$ generators, clarifying their algebraic structure and connections.

Findings

Reduction of zero curvature to Lax equation with $W_{ty}$ generators

Clarification of relation between Toda theories and KP hierarchy

Discussion of potential matrix model correspondence

Abstract

We reexamine the $W_{\infty}$ symmetry of the $sl(N)$ Conformal Affine Toda theories. It is shown that it is possible to reduce (nonuniquely) the zero curvature equation to a Lax equation for a first order pseudodifferential oprator, whose coefficients are the generators of the $W_{\infty}$ algebra. This clarifies the known relation between the Conformal Affine Toda theories and the KP hierarchy. A possible correspondence between the matrix models and the Conformal Affine Toda models is discussed.