Elliptic Calogero–Moser systems and isomonodromic deformations

Abstract

We show that various models of the elliptic Calogero–Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a nonautonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero–Moser system turns out to give a Lax representation of the nonautonomous system as well. This Lax representation ensures that the nonautonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the “extended twisted BCl model” recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev’s generalized elliptic Calogero–Moser system. We use the “root-type” Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.