Lax equations for relativistic ${\rm GL}(NM,{\mathbb C})$ Gaudin models on elliptic curve

TL;DR

This paper classifies and describes the most general relativistic elliptic Gaudin models on GL(NM) bundles, unifying various known models and providing explicit parametrizations and R-matrix formulations.

Contribution

It introduces the most general GL(NM) elliptic integrable system with a Lax matrix having multiple poles, unifying several known models and extending the theory of relativistic Gaudin models.

Findings

Classified the general GL(NM) elliptic integrable system.

Connected the model to inhomogeneous Ruijsenaars chain.

Provided explicit parametrization of spin variables.

Abstract

We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ interacting relativistic ${\rm GL}_N$ tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on ${\rm GL}$-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars-Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general ${\rm GL}_{NM}$ model is also described through $R$-matrices satisfying associative Yang-Baxter equation. This description provides the trigonometric and rational analogues of ${\rm GL}_{NM}$ models.