Moduli spaces for linear differential equations and the Painlev\'e equations

TL;DR

This paper systematically constructs and classifies ten families of rank two connections on the projective line that induce all Painlevé equations, by analyzing their monodromy data and explicit equations.

Contribution

It provides a comprehensive classification of isomonodromic families related to Painlevé equations using moduli space and monodromy space analysis.

Findings

Explicit equations for monodromy spaces of ten families.

Natural explicit families of connections constructed.

Derivation of all Painlevé equations from these families.

Abstract

In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlev\'e equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit families of connections for these ten cases and calculate isomonodromic equations which give Painlev\'e equations of all types.