A classical variant of the vertex algebra & the auxiliary linear problem

TL;DR

This paper introduces a classical analogue of vertex algebra for integrable field theories, linking auxiliary functions to classical vertex operators and identifying local integrals of motion.

Contribution

It develops a classical vertex algebra framework to describe auxiliary functions and integrals of motion in classical integrable systems, including systems with defects.

Findings

Classical vertex operators encode auxiliary functions.

Local integrals of motion are in involution.

Framework applies to systems with defects and on semi-infinite lines.

Abstract

We propose a classical analogue of the vertex algebra in the context of classical integrable field theories. We use this fundamental notion to describe the auxiliary function of the linear auxiliary problem as a classical vertex operator. Then using the underlying algebra satisfied by the auxiliary function together with the linear auxiliary problem we identify the local integrals of motion, which by construction are in involution. The time components of the Lax pair are also identified in terms of the classical vertex operators. Systems in the presence of point like defects as well as systems on the semi-infinite line are investigated. Specific examples associated to the classical Yangian and twisted Yangian are also presented.