Dimers and cluster integrable systems

TL;DR

This paper demonstrates that dimer models on bipartite graphs on a torus naturally give rise to quantum integrable systems, connecting statistical mechanics, algebraic geometry, and network theory.

Contribution

It introduces a novel link between dimer models and cluster integrable systems, revealing the phase space structure and associated Lagrangian subvarieties.

Findings

The phase space includes the moduli space of line bundles with connections.

The sum of Hamiltonians corresponds to the dimer partition function.

Constructs discrete quantum integrable systems from bipartite graphs.

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.