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 $N$.

arXiv commentAdded a background section 2.4, and a new section 6.1. Appendix is now section 5 and contains more details

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