Field analogue of the Ruijsenaars-Schneider 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 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 → 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.

Lax equations for relativistic GL(NM,C) Gaudin models on elliptic curve

“…For example, the models 8 and 9 are gauge equivalent. This phenomenon was originally observed by Hasegawa [15], see also [8,17,41,43].…”

“…Gauge equivalence between RS model and relativistic top. Here, following [15] (see also [8,17,41,43]) we briefly describe the change of variables between the models 8 and 9 on the scheme 2.…”

“…with some matrix S (see [8,15,41] and the appendix in [43] for details). The matrix S in this case is special.…”

“…The Ruijsenaars chain on n sites is the model introduced recently in [43]. Similarly to (2.54) it is described by N × N monodromy matrix…”

“…At the relativistic level the 1 + 1 field theories corresponding to the models 5 on the scheme 2 are known from [9]. Recently the 1 + 1 analogue of the Ruijsenaars-Schneider model was suggested in [43]. It is an interesting classification problem to describe the soliton equations related to all boxes on the schemes 1 and 2.…”