$q$-Painlev\'e equations on cluster Poisson varieties via toric geometry

TL;DR

This paper connects $q$-Painlevé equations with cluster Poisson varieties using toric geometry, classifies their seeds, and realizes these systems as automorphisms on associated cluster varieties.

Contribution

It introduces the notion of $q$-Painlevé type seeds, classifies them, and links $q$-Painlevé equations to cluster Poisson varieties through toric models.

Findings

Classification of $q$-Painlevé seeds matches Sakai's classification.

Realization of $q$-Painlevé systems as automorphisms on cluster varieties.

Establishment of a geometric framework connecting $q$-Painlevé equations and cluster Poisson varieties.

Abstract

We provide a relation between the geometric framework for $q$-Painlev\'{e} equations and cluster Poisson varieties by using toric models of rational surfaces associated with $q$-Painlev\'{e} equations. We introduce the notion of seeds of $q$-Painlev\'{e} 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 $q$-Painlev\'{e} equations given by Sakai. We realize $q$-Painlev\'{e} systems as automorphisms on cluster Poisson varieties associated with seeds of $q$-Painlev\'{e} type.