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.

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

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• Spanning forests and the vector bundle Laplacian
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• Cluster algebras I: Foundations
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• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
• New Directions in Bipartite Field Theories
• Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
• 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