Dimer models and the special McKay correspondence

TL;DR

This paper explores the relationship between dimer models and the special McKay correspondence, showing how modifications of lattice polygons relate to derived categories of coherent sheaves on toric Calabi-Yau 3-folds.

Contribution

It introduces a refined operation on dimer models linked to the McKay correspondence, establishing a correspondence with derived categories of toric Calabi-Yau 3-folds.

Findings

Refined operation on lattice polygons via corner removal.

Equivalence of derived categories for dimer models and toric Calabi-Yau 3-folds.

Independent proof not relying on previous results.

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 relationship between combinatorics of dimer models and geometry of moduli spaces, and does not depend on the result of math/9908027.