Differential Galois Theory and Isomonodromic Deformations

TL;DR

This paper develops a geometric framework for differential Galois theory of parameter-dependent connections, linking Galois groups to isomonodromic deformations and applying it to classical special functions.

Contribution

It introduces a geometric setting for the differential Galois theory of G-invariant connections with parameters and relates Galois groups to isomonodromic deformations.

Findings

Galois group determination via isomonodromic deformations

Computation of Galois groups for Fuchsian systems

Analysis of Gauss hypergeometric equation

Abstract

We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.