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Cluster Reductions, Mutations, and $q$-Painlevé Equations

Mikhail Bershtein, Pavlo Gavrylenko, Andrei Marshakov, Mykola Semenyakin

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 Painlevé 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 Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group.

Cluster Reductions, Mutations, and $q$-Painlevé Equations

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 -difference Painlevé 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 Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group.

Paper Structure

This paper contains 36 sections, 40 theorems, 129 equations, 56 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Acknowledgment
  4. Dimer models and Goncharov-Kenyon integrable systems
  5. Consistent bipartite graphs
  6. Dimer models
  7. Poisson structure. Integrability
  8. Cluster mutations and face mutations
  9. Zigzag mutations and reductions
  10. Length 4 zigzags
  11. Polynomial and polygon mutations
  12. Zigzag mutations 1: bipartite graphs
  13. Zigzag mutations 2: dimer models
  14. Zigzag mutations 3: Poisson geometry
  15. Example: length 6 zigzags
  16. ...and 21 more sections

Key Result

Theorem 2.6

Let $D_0$ be a dimer cover and $\operatorname{sgn}_K$ be a Kasteleyn sign. Then where $q_{K,D_0}$ is a quadratic form. Moreover, the map $K \mapsto q_{K,D_0}$ is a one to one correspondence between the set of equivalence classes of Kasteleyn signs and set of quadratic forms, this correspondence depends on choice of $D_0$.

Figures (56)

  • Figure 1.1: On the left decorated polygon, on the right corresponding spectral curve.
  • Figure 1.2: On the left face mutation, on the right zigzag mutation, drawn on the cylinder
  • Figure 2.1: Zigzag path
  • Figure 2.2: A piece of bipartite graph $\Gamma$ and a corresponding piece of the quiver $\mathcal{Q}$ (in blue).
  • Figure 2.3: 4-gon face mutation (spider move)
  • ...and 51 more figures

Theorems & Definitions (99)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Kasteleyn:1963, Cimasoni:2007dimers
  • Theorem 2.7: George2022inverse, Broomhead:2010
  • Definition 2.8
  • Theorem 2.9: Gulotta:2008properly, Goncharov:2013
  • Theorem 2.10: Goncharov:2013
  • Remark 2.11
  • ...and 89 more