On the geometry of isomonodromic deformations on the torus and the elliptic Calogero-Moser system

TL;DR

This paper explores the geometric structure of isomonodromic deformations on the torus, linking them to the elliptic Calogero-Moser system, and introduces an extended symplectic form demonstrating its closure.

Contribution

It extends the understanding of isomonodromic deformations on the torus by revealing new symmetries and integrating the elliptic Calogero-Moser system into the geometric framework.

Findings

Extended symmetry for isomonodromic deformations established

Elliptic Calogero-Moser system integrated into the geometric framework

Extended symplectic form shown to be closed

Abstract

We consider isomonodromic deformations of connections with a simple pole on the torus, motivated by the elliptic version of the sixth Painlev\'e equation. We establish an extended symmetry, complementing known results. The Calogero-Moser system in its elliptic version is shown to fit nicely in the geometric framework, the extended symplectic two-form is introduced and shown to be closed.