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

Autor:	Bershtein, Mikhail, Gavrylenko, Pavlo, Marshakov, Andrei, Semenyakin, Mykola
Rok vydání:	2024
Předmět:	Nonlinear Sciences - Exactly Solvable and Integrable Systems High Energy Physics - Theory Mathematical Physics Mathematics - Quantum Algebra
Druh dokumentu:	Working Paper
Popis:	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. Comment: 60 pages, 56 figures, preliminary version
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2411.00325 Zobrazit plný text záznamu View this record from Arxiv