Painlev\'{e} type equations and Hitchin systems

TL;DR

This survey explores the connection between Painlevé equations, Hitchin systems, and isomonodromy preserving equations on Riemann surfaces, highlighting their Hamiltonian structure and relations to quantization and classical limits.

Contribution

It provides a unified interpretation of isomonodromy equations as reduced Hamiltonian systems linked to Hitchin moduli spaces and discusses their relation to quantization and classical limits.

Findings

Relation of Painlevé VI to Hitchin systems

Connection between isomonodromy equations and Whitham quantization

Classical limit of KZB equations linked to these systems

Abstract

In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles in the marked points. We discuss relations of these equations with the Whitham quantization of the Hitchin systems and with the classical limit of the Knizhnik-Zamolodchikov-Bernard equations. The main example is the one-parameter family of Painlev\'{e} VI equation and its multicomponent generalization.