Universal Schlesinger system and Birkhoff factorization

Abstract

The aim of this paper is to establish an infinite dimensional generalization of the Schlesinger system -- a system of PDE's describing isomonodromic deformations of Fuchsian systems. This universal Schlesinger system first appeared in a paper by Korotkin and Samtleben in the finite dimensional case (i.e. when it reduces to the finite dimensional Schlesinger system). We intend here to establish it in its full generality, and to give its geometrical meaning in terms of deformations of Birkhoff factorizations. The Birkhoff factorization we are considering consists in finding a piecewise holomorphic matrix-valued function $Y$ on $\mathbb P^1\smallsetminus\mathbb S^1$ which admits a prescribed multiplicative jump $G : \mathbb S^1\to \text{GL}_N(\mathbb C)$ across the unit circle $\mathbb S^1$. We explain how this factorization can be deformed by the action of the group of diffeomorphisms of the circle, which acts by composition on the multiplicative jump $G$. The integrability equations of this "isomonodromic" deformation is given by the universal Schlesinger system, which is then naturally expressed by means of Virasoro generators.