On the space of $2d$ integrable models

TL;DR

This paper explores the symmetry structures of 2D integrable models through infinite-dimensional Lie algebras, identifying known and new commuting charges, and examining their deformations upon quantization.

Contribution

It introduces a framework linking Lie algebras to 2D integrable models, discovering new sequences of commuting charges and analyzing their quantum deformations.

Findings

Identified symmetry algebras of known integrable models.

Discovered new sequences of commuting charges.

Showed Lie algebra deformations upon quantization.

Abstract

We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We study the commuting subalgebras of the $2d$ free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.