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Stokes manifolds and cluster algebras

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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03225645
Contributor : Sofia Tarricone Connect in order to contact the contributor
Submitted on : Wednesday, May 12, 2021 - 6:07:06 PM
Last modification on : Wednesday, October 20, 2021 - 3:19:02 AM

Identifiers

• HAL Id : hal-03225645, version 1
• ARXIV : 2104.13784

Citation

Marco Bertola, Sofia Tarricone. Stokes manifolds and cluster algebras. 2021. ⟨hal-03225645⟩

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