# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.01636