Cluster Reductions, Mutations, and $q$-Painlev\'e Equations

TL;DR

This paper extends cluster integrable systems via Hamiltonian reductions to encompass all $q$-Painlevé equations, revealing dual cluster structures and symmetries that unify spectral curves and Painlevé Hamiltonians.

Contribution

It introduces a novel extension of Goncharov-Kenyon systems through reductions, filling the gap in cluster construction of $q$-Painlevé equations and uncovering dual cluster structures and symmetries.

Findings

All $q$-Painlevé equations are obtained as deautonomizations of reduced Goncharov-Kenyon systems.

Isomorphisms of reduced systems are given by polynomial and polygon mutations.

The approach reveals a self-duality and extended symmetry between spectral curves and Painlevé Hamiltonians.

Abstract

We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlev\'e equations, showing that all of them can be obtained as deautonomizations of the reduced Goncharov-Kenyon systems. Conjecturally, the isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in some sense, cluster structure. These are the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons. In the Painlev\'e case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlev\'e Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group.