Generalized NLS Hierarchies from Rational $W$ Algebras

TL;DR

This paper explores the connection between rational $ ext{W}$ algebras and integrable hierarchies, demonstrating how specific coset constructions relate to known integrable systems like the KP and Non-Linear Schrödinger hierarchies.

Contribution

It establishes a method to associate rational $ ext{W}$ algebras with integrable hierarchies and provides explicit examples linking coset algebras to well-known integrable systems.

Findings

Rational $ ext{W}$ algebras can be related to integrable hierarchies.

Explicit examples include the Non-Linear Schrödinger and KP hierarchies.

Rational algebras emerge as constraints in hierarchy reductions.

Abstract

Finite rational $\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational $\cw$ algebra its corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset, leading to the Non-Linear Schr\"{o}dinger hierarchy, and the $U(1)$ coset of the Polyakov-Bershadsky $\cw$ algebra, leading to a $3$-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies.