Geometric Reid's recipe for dimer models

Understand

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.