Geometric Reid’s recipe for dimer models

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 $$\mathbb {C}[x,y,z]*G$$ for a finite abelian subgroup of $$\mathrm{SL }(3,\mathbb {C})$$ . 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 (J Reine Angew Math 636:193–236, 2009) and Cautis–Craw–Logvinenko (J Reine Angew Math arXiv:1205.3110 , 2014) 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 (J Algebra 324:2064–2087, 2010).