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Dimers and cluster integrable systems

A. B. Goncharov, R. Kenyon

TL;DR

We establish a Poisson-geometric framework for dimers on graphs embedded on surfaces and demonstrate that when the graph lies on a torus, the resulting moduli space of line bundles with connections yields a cluster Poisson variety with a family of commuting Hamiltonians, giving a classical integrable system whose partition function expresses the Hamiltonians. The work develops a quantum deformation via a quantum torus, producing commuting quantum Hamiltonians and a discrete cluster evolution (octahedron/cube recurrences) that preserves the spectrum. By embedding dimers into cluster varieties and relating to resistor networks, the authors identify a Lagrangian resistor-network subvariety and show its cluster nature, enabling a unified algebraic-geometric description through spectral data and Beauville-type integrable systems. The paper also forges connections to Teichmüller theory, outlines discrete integrable systems on torus grids, and situates these constructions within toric geometry and Calabi–Yau perspectives, suggesting broad applicability to mathematical physics and integrable systems. Overall, it provides a comprehensive framework linking dimer models, cluster algebras, spectral data, and discrete and continuous integrable dynamics.

Abstract

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.

Dimers and cluster integrable systems

TL;DR

We establish a Poisson-geometric framework for dimers on graphs embedded on surfaces and demonstrate that when the graph lies on a torus, the resulting moduli space of line bundles with connections yields a cluster Poisson variety with a family of commuting Hamiltonians, giving a classical integrable system whose partition function expresses the Hamiltonians. The work develops a quantum deformation via a quantum torus, producing commuting quantum Hamiltonians and a discrete cluster evolution (octahedron/cube recurrences) that preserves the spectrum. By embedding dimers into cluster varieties and relating to resistor networks, the authors identify a Lagrangian resistor-network subvariety and show its cluster nature, enabling a unified algebraic-geometric description through spectral data and Beauville-type integrable systems. The paper also forges connections to Teichmüller theory, outlines discrete integrable systems on torus grids, and situates these constructions within toric geometry and Calabi–Yau perspectives, suggesting broad applicability to mathematical physics and integrable systems. Overall, it provides a comprehensive framework linking dimer models, cluster algebras, spectral data, and discrete and continuous integrable dynamics.

Abstract

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.

Paper Structure

This paper contains 88 sections, 45 theorems, 160 equations, 43 figures.

Table of Contents

  1. Introduction
  2. Dimer models and Poisson geometry.
  3. Conjugate surface graph and Poisson structure
  4. Natural coordinates on ${{\mathcal{L}}}_{\Gamma}$.
  5. Zig-zag loops and the center of the Poisson algebra of functions on ${\mathcal{L}}_\Gamma$.
  6. Gluing the Poisson phase space
  7. Cluster integrable systems for dimer models on a torus.
  8. Classical integrable system
  9. Quantum integrable system
  10. Discrete cluster integrable systems.
  11. Cluster nature of a resistor network model on a torus
  12. Dimer model arising from a graph on a torus
  13. Resistor network Lagrangian subvariety
  14. Toric surfaces, spectral data and the Beauville integrable system
  15. The Kasteleyn operator
  16. ...and 73 more sections

Key Result

Lemma 1.1

Let $Z$ be an (oriented) zig-zag loop on a bipartite oriented surface graph $\Gamma$. Then as $Z$ runs over zig-zag loops, the functions $W_Z$ generate the center of the Poisson algebra ${\mathcal{O}}({{\mathcal{L}}}_\Gamma)$ of functions on ${{\mathcal{L}}}_\Gamma$. The product of all of them is $1

Figures (43)

  • Figure 1: A Zig-zag path on a ribbon graph.
  • Figure 2: The bipartite graph $\Gamma_G$ (dashed and solid edges) assigned to a graph $G$ (solid edges).
  • Figure 3: Zig-zag strands and surrounding $4$-gon.
  • Figure 4: Minimal bipartite surface graphs have no self-intersections and parallel bigons (left); antiparallel bigons (right) are allowed.
  • Figure 5: The $2$ - $2$ move.
  • ...and 38 more figures

Theorems & Definitions (88)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Thurston
  • Lemma 2.4
  • proof
  • ...and 78 more