On the Geometry of Isomonodromic Deformations

TL;DR

This paper explores the geometric and Hamiltonian structure of isomonodromic deformations of connections on vector bundles over Riemann surfaces, highlighting their relation to moduli spaces and twists supported over points.

Contribution

It provides a geometric interpretation of the Hamiltonian structure of isomonodromic deformations using moduli of pairs and twists, offering new insights into their underlying geometry.

Findings

Identifies the moduli of pairs as generated by twists over points.

Shows the Hamiltonian nature of the difference between deformations.

Connects isomonodromic deformations to geometric structures on vector bundles.

Abstract

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs $(V,\nabla)$ of vector bundles and connections as being obtained by "twists" supported over points of a fixed vector bundle $V_0$ with a fixed connection $\nabla_0$; this gives two deformations, one, isomonodromic, of $(V,\nabla)$, and another induced from the isomonodromic deformation of $(V_0,\nabla_0)$. The difference between the two will be Hamiltonian.