Stokes Manifolds and Cluster Algebras

Stokes’ manifolds, also known as wild character varieties, carry a natural Poisson structure. Our goal is to provide explicit log-canonical coordinates for this Poisson structure on the Stokes’ manifolds of polynomial connections of rank 2, thus including the second Painlevé hierarchy. This construction provides the explicit linearization of the Poisson structure first discovered by Flaschka and Newell and then rediscovered and generalized by Boalch. We show that, for a connection of degree K , the Stokes’ manifold is a cluster manifold of type A 2 K with one frozen vertex. The main idea is then applied to express explicitly also the log–canonical coordinates for the Poisson bracket introduced by Ugaglia in the context of Frobenius manifolds and then also applied by Bondal in the study of the symplectic groupoid of quadratic forms.