Field analogue of the Ruijsenaars-Schneider model

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 -> 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.

Automated summary

This article proposes a field extension of the classical elliptic Ruijsenaars-Schneider model and defines and derives it through two different methods. The first method defines the model through the trace of the L-matrix, resulting in a lattice field analogue. The second method defines the model through the investigation of elliptic families of solutions to the 2D Toda equation and proves that their equations of motion are Hamiltonian. The models obtained from these two methods are equivalent.