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 Painleve differential equations. Since all Painleve 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 Teichmuller 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 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 Painleve differential equations in geometric terms.