Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations

TL;DR

This paper explores the connection between Calogero-Moser-Sutherland models and Landau-Lifshitz equations through a detailed study of the IRF-Vertex relation, Higgs bundles, and gauge equivalences, focusing on the ${ m sl}_2$ case.

Contribution

It provides a detailed construction of the continuous IRF-Vertex relation and establishes gauge equivalences between integrable models and field theories for the ${ m sl}_2$ case.

Findings

Explicit variable transformations between rational, trigonometric, and elliptic models.

Description of gauge equivalence between L-A pairs of Landau-Lifshitz equations and Calogero-Moser-Sutherland models.

Construction of Higgs bundles of infinite rank over elliptic curves.

Abstract

We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence between L-A pairs of the Landau-Lifshitz type equations and 1+1 field theory generalization of the Calogero-Moser-Sutherland models is described. In this paper the ${\rm sl}_2$ case is studied. Explicit changes of variables are obtained between the rational, trigonometric and elliptic models.