Crepant resolutions and brane tilings I: Toric realization

TL;DR

This paper explores the relationship between brane tilings and toric Calabi-Yau varieties, providing explicit descriptions of crepant resolutions and linking the McKay correspondence to brane tilings.

Contribution

It offers a detailed toric description of all commutative crepant resolutions associated with brane tilings and connects these to the McKay correspondence in three dimensions.

Findings

Explicit toric descriptions of crepant resolutions

Interpretation of McKay correspondence via brane tilings

Brane tilings encode resolutions of singular Calabi-Yau varieties

Abstract

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of all its commutative crepant resolutions. We also explain how the McKay correspondence in dimension 3 can be interpreted using brane tilings.