Dimer models and Calabi-Yau algebras

TL;DR

This paper explores the relationship between dimer models and Calabi-Yau algebras, establishing conditions under which these models produce 3D Calabi-Yau algebras and non-commutative resolutions of certain toric varieties.

Contribution

It proves that algebraically consistent dimer models yield 3D Calabi-Yau algebras and serve as non-commutative crepant resolutions of Gorenstein affine toric threefolds.

Findings

Geometrically consistent models are algebraically consistent.

Algebraically consistent dimer models produce 3D Calabi-Yau algebras.

These algebras provide non-commutative crepant resolutions.

Abstract

In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. We prove that the algebra obtained from an algebraically consistent dimer model is a 3-dimensional Calabi-Yau algebra and finally prove that this gives a non-commutative crepant resolution of the Gorenstein affine toric threefold associated to the dimer model.