Consistency conditions for dimer models

TL;DR

This paper investigates various consistency conditions for dimer models on a torus, proving their equivalence and clarifying when these models yield noncommutative crepant resolutions of toric Gorenstein singularities.

Contribution

It provides a detailed comparison of different consistency notions for dimer models and establishes their equivalence on a torus, enhancing understanding of their role in algebraic geometry.

Findings

All major consistency notions for dimer models on a torus are equivalent.

Consistency conditions determine when dimer models produce noncommutative crepant resolutions.

Clarifies the relationship between combinatorial properties and algebraic resolutions.

Abstract

Dimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper we study the major different notions in detail and show that for dimer models on a torus they are all equivalent.