Deformations of Dimer Models

TL;DR

This paper introduces deformations of consistent dimer models that realize combinatorial mutations of their associated perfect matching polygons, linking dimer model transformations to mirror symmetry and toric geometry.

Contribution

It defines a new set of operations called deformations of consistent dimer models that correspond to polygon mutations, connecting combinatorial and geometric frameworks.

Findings

Deformations of dimer models realize polygon mutations.

Dimer model deformations preserve consistency conditions.

Link established between dimer model transformations and mirror symmetry.

Abstract

The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon $P$ there exists a dimer model having $P$ as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.