Moduli Spaces for the Fifth Painlev\'e Equation

TL;DR

This paper explores the geometric and algebraic structures underlying the fifth Painlevé equation, including moduli spaces, monodromy, and Lax pairs, providing explicit formulas and a polynomial Hamiltonian.

Contribution

It introduces a natural moduli space of rank 4 connections that induces the Noumi-Yamada Lax pair for P5, and derives a polynomial Hamiltonian equivalent to Okamoto's.

Findings

Explicit formulas for Stokes matrices and parabolic structures.

The rank 4 Lax pair is induced by a natural moduli space.

A polynomial Hamiltonian for P5 is obtained.

Abstract

Isomonodromy for the fifth Painlev\'e equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlev\'e spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.