The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

TL;DR

This paper presents a new method to describe pairs of bundles with logarithmic connections on Riemann surfaces using a surface presentation as a disc exterior, leading to a modified Schlesinger system for isomonodromic deformations.

Contribution

It introduces a novel presentation of bundles with connections on Riemann surfaces and derives a modified Schlesinger system describing their isomonodromic deformations.

Findings

Derived a modified Schlesinger system for Riemann surfaces.

Presented a new surface presentation approach for bundles with connections.

Connected the deformation equations to classical Schlesinger systems.

Abstract

We introduce a way of presentation of pairs $(E,\nabla)$, where $E$ is a bundle on a Riemann surface and $\nabla$ is a logarithmic connection in $E$, which is based on a presentation of the surface as a factor of the exterior of the unit disc. In this presentation we write the local equation of isomonodormic deformation of pairs $(E,\nabla)$. These conditions are written as a modified Schlesinger system on a Riemann sphere (and in the typical case just as an ordinary Schlesinger system) plus some linear system.