Geometric Reid's recipe for dimer models

TL;DR

This paper explores a geometric and algebraic framework connecting crepant resolutions of toric Gorenstein singularities with dimer models, extending known derived equivalences and describing the behavior of certain sheaves.

Contribution

It generalizes the Fourier-Mukai transform for crepant resolutions to the dimer model setting and describes the images of simple modules under this equivalence.

Findings

The derived equivalence sends vertex simples to pure sheaves.

Except for the zero vertex, which maps to the dualising complex.

Explicit descriptions and support computations of these sheaves.

Abstract

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution $Y$, this derived equivalence generalises the Fourier-Mukai transform relating the $G$-Hilbert scheme and the skew group algebra $\CC[x,y,z]\ast G$ for a finite abelian subgroup of $\SL(3,\CC)$. We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis-Logvinenko and Cautis-Craw-Logvinenko to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko.