Dimers and cluster integrable systems

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.

This is an updated version, 75 pages, which will appear in Ann. Sci. ENS

References in corpus (3)

• Total positivity, Grassmannians, and networks
• Planar dimers and Harnack curves
• Quantum cluster algebras

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• A 5d/3d duality from relativistic integrable system
• Bipartite Field Theories from D-Branes
• Scattering Amplitudes and Toric Geometry
• Cluster Transformations from Bipartite Field Theories
• T-systems, networks and dimers
• Discrete Painlevé equations from Y-systems
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups