2011/07/27 by A. B. Goncharov, Richard Kenyon, Goncharov, A. B. +1 · 6 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.1107.5588
openalex publication_date 2011/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.