Field analogue of the Ruijsenaars-Schneider model

TL;DR

This paper introduces a field extension of the elliptic Ruijsenaars-Schneider model, connecting it to elliptic spin chains and the 2D Toda equation, providing new insights into integrable systems with lattice and continuum limits.

Contribution

It presents a novel field analogue of the Ruijsenaars-Schneider model derived through two methods, linking it to elliptic spin chains and 2D Toda solutions, and explores its continuum and discrete limits.

Findings

Model is gauge equivalent to an elliptic spin chain.

Derived equations of motion as difference equations with a Hamiltonian structure.

Limit $ o 0$ recovers the field extension of the Calogero-Moser model.

Abstract

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of $L$-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing $\eta$, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit $\eta \to 0$ is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed.