Stokes manifolds and cluster algebras

TL;DR

This paper provides explicit log-canonical coordinates for the Poisson structures on Stokes' manifolds associated with polynomial connections of rank 2, revealing their cluster algebra structure and linearizing the Poisson brackets.

Contribution

It explicitly constructs log-canonical coordinates for the Poisson structures on Stokes' manifolds, showing they are cluster manifolds of type A_{2K} and generalizing previous work.

Findings

Stokes' manifolds are cluster manifolds of type A_{2K}.

Explicit log-canonical coordinates are provided for the Poisson structures.

The construction generalizes and linearizes known Poisson brackets.

Abstract

Stokes' manifolds, also known as wild character varieties, carry a natural symplectic structure. Our goal is to provide explicit log-canonical coordinates for these natural Poisson structures on the Stokes' manifolds of polynomial connections of rank $2$, thus including the second Painlev\'e\ 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 $A_{2K}$. 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.