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Dimer models and the special McKay correspondence

Akira Ishii and Kazushi Ueda

Geometry & Topology 19 (2015) 3405–3466

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.

dimer model, McKay correspondence, derived equivalence
Mathematical Subject Classification 2010
Primary: 14F05
Secondary: 14D20, 16G20, 14E16
Received: 12 June 2014
Revised: 25 September 2014
Accepted: 1 April 2015
Published: 6 January 2016
Proposed: Lothar Göttsche
Seconded: Richard Thomas, Jim Bryan
Akira Ishii
Division of Mathematical and Information Sciences
Hiroshima University
1-7-1 Kagamiyama
Higashi-Hiroshima 739-8521
Kazushi Ueda
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1 Komaba
Meguro-ku 153-8914