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A cluster ensemble is a pair of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group . The space is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism . The space is equipped with a closed -form, possibly degenerate, and the space has a Poisson structure. The map is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type, which we call a. 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.
We say that two such graphs and areif the Newton polygons of the corresponding partition functions coincide up to translation. We define...
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