Painlev\'e monodromy manifolds, decorated character varieties and cluster algebras

TL;DR

This paper introduces decorated character varieties for Riemann surfaces related to Painlevé equations, revealing their Poisson structure and cluster algebra connections, and geometrically describing confluence procedures.

Contribution

It defines decorated character varieties for Painlevé-related surfaces, computes their Poisson brackets, and links confluence of Painlevé equations to geometric transformations.

Findings

Decorated character varieties form Poisson manifolds with cluster algebra structures.

Explicit Poisson brackets are computed for these varieties.

Geometric interpretation of Painlevé confluence procedures is provided.

Abstract

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlev\'e differential equations. Since all Painlev\'e differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichm\"uller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimension 3 s+ 2 n-6 and we explicitly compute the Poisson brackets which are naturally of cluster type. We also show how to obtain the confluence procedure of the Painlev\'e differential equations in geometric terms.