Field analogue of the Ruijsenaars-Schneider model

TL;DR

This paper constructs a field-theoretic generalization of the elliptic Ruijsenaars-Schneider model in 1+1 dimensions by presenting two equivalent formulations: (i) a finite Ruijsenaars-Schneider chain obtained as a gauge-equivalent reduction of a classical elliptic ${\rm GL}_N$ spin chain, and (ii) a field system arising from elliptic families of solutions to the 2D Toda lattice, whose pole dynamics yield RS-type equations. The authors establish a coherent Lax/zero-curvature framework for both approaches, identify the underlying Hamiltonian structures, and show that the lattice field theory reduces to the RS chain under discretization, while the $\eta\to0$ limit recovers the Calogero-Moser field theory studied in AKV02. A fully discrete version is also constructed via Hirota-based elliptic families, and the work connects IRF-Vertex, Sklyanin algebras, and Lax representations to a new integrable field theory with potential links to Landau-Lifshitz-type models. Overall, the paper extends the RS paradigm to continuous-field settings, enriching the landscape of integrable systems and offering new avenues for connections to lattice gauge theories and algebraic geometry. The results provide explicit Lax pairs, $M$-matrices, and Hamiltonians that underlie the pole dynamics and their field-theoretic realizations, highlighting the deep interplay between elliptic $R$-matrices, IRF-Vertex dualities, and multi-pole reductions. The field RS model thus serves as a natural field extension of the RS–Calogero correspondence with significant structural and conceptual implications for integrable hierarchies.

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 $η$, 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 $η\to 0$ 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.