The cluster modular group of the dimer model

Abstract

Associated to a convex integral polygon $N$ is a cluster integrable system $\mathcal X\_N$ constructed from the dimer model. We compute the group $G\_N$ of symmetries of $\mathcal X\_N$, called the (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of $G\_N$ are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, $G\_N$ is a subgroup of the Picard group of a certain algebraic surface associated to $N$.