Painlevé monodromy manifolds, decorated character varieties and cluster algebras

Abstract

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé 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 spaceintroduced in [8]. We show that the decorated character variety of a Riemann sphere withs holes and n >1 cusps is a Poisson manifold of dimension 3s+ 2n−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 Painlevé differential equations in geometric terms