Dimer models and the special McKay correspondence

Abstract

We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon there is a dimer model such that the derived category of finitely generated modules over the path algebra of the corresponding quiver with relations is equivalent to the derived category of coherent sheaves on a toric Calabi–Yau 3–fold determined by the lattice polygon. Our proof is based on a detailed study of the relationship between combinatorics of dimer models and geometry of moduli spaces, and does not depend on the result of Bridgeland, King and Reid.