# Local charges in involution and hierarchies in integrable sigma-models

## TL;DR

This paper constructs an infinite set of local conserved charges in involution for a broad class of integrable sigma-models, revealing a hierarchy of compatible integrable equations linked to the algebraic structure of the models.

## Contribution

It extends the construction of local charges in involution to a wide class of non-ultralocal integrable sigma-models with classical Lie algebras, generalizing previous approaches.

## Key findings

- Established local charges attached to zeros of the twist function.
- Showed charges generate an infinite hierarchy of integrable flows.
- Connected charges to the exponents of affine Kac-Moody algebras.

## Abstract

Integrable $\sigma$-models, such as the principal chiral model, ${\mathbb{Z}}_T$-coset models for $T \in {\mathbb{Z}}_{\geq 2}$ and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) $r/s$-systems with twist function. In this general setting, and when the Lie algebra ${\mathfrak{g}}$ underlying the $r/s$-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space $\sigma$-model. In the present context, the local charges are attached to certain `regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra $\widehat{{\mathfrak{g}}}$ associated with ${\mathfrak{g}}$. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.