$q$-Painlevé equations on cluster Poisson varieties via toric geometry

Abstract

We provide a relation between the geometric framework for $q$-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with $q$-Painlevé equations. We introduce the notion of seeds of $q$-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of $q$-Painlevé equations given by Sakai. We realize $q$-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of $q$-Painlevé type.

28pages, v2: add Section 3.1, v3, v4: minor changes

References in corpus (3)

• Moduli of surfaces with an anti-canonical cycle
• Cluster integrable systems, q-Painleve equations and their quantization
• Cluster Automorphisms