On the relation between integrability and infinite-dimensional algebras

TL;DR

This paper explores the deep connection between integrability in physical systems and the structure of infinite-dimensional algebras, revealing how certain algebraic symmetries underpin integrable models like Liouville and NLS equations.

Contribution

It demonstrates the construction of commuting charges from Kac-Moody currents, relates W-algebras to Liouville theory, and links the NLS equation with the \\Wi algebra, proposing a fundamental link between integrability and algebraic structures.

Findings

Existence of sets of commuting charges from U(1) Kac-Moody currents.

W-algebras as symmetries of Liouville theory at special couplings.

Relationship between NLS equation, KP hierarchy, and \\Wi algebra.

Abstract

We review our work on the relation between integrability and infinite-dimensional algebras. We first consider the question of what sets of commuting charges can be constructed from the current of a \mbox{\sf U}(1) Kac-Moody algebra. It emerges that there exists a set $S_n$ of such charges for each positive integer $n>1$; the corresponding value of the central charge in the Feigin-Fuchs realization of the stress tensor is $c=13-6n-6/n$. The charges in each series can be written in terms of the generators of an exceptional \W-algebra. We show that the \W-algebras that arise in this way are symmetries of Liouville theory for special values of the coupling. We then exhibit a relationship between the \nls equation and the KP hierarchy. From this it follows that there is a relationship between the \nls equation and the algebra \Wi. These examples provide evidence for our conjecture that the phenomenon of integrability is intimately linked with properties of infinite dimensional algebras.