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 Teichmüller 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 Painlevé differential equations in geometric terms.

Parts of this paper overlap with arXiv:1212.6723, but it presents a much deeper geometric understanding and the new notion of decorated character variety which was not included there. 19 coloured pictures, 33 pages. To appear on Int. Mat. Res. Not