Painlevé monodromy manifolds, decorated character varieties, and cluster algebras

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, we show that it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character variety is considered here as complexification of the bordered cusped Teichmüller space introduced in [11]. 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 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.