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Dimer models and the
special McKay correspondence
Akira Ishii and Kazushi Ueda |
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Geometry & Topology 19 (2015) 3405–3466
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Abstract |
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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. |
Keywords
dimer model, McKay correspondence, derived equivalence
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Mathematical Subject Classification 2010
Primary: 14F05
Secondary: 14D20, 16G20, 14E16
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References |
Publication
Received: 12 June 2014
Revised: 25 September 2014
Accepted: 1 April 2015
Published: 6 January 2016
Proposed: Lothar Göttsche
Seconded: Richard Thomas, Jim Bryan
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Authors |
| Akira Ishii | |
| Division of Mathematical and
Information Sciences Hiroshima University 1-7-1 Kagamiyama Higashi-Hiroshima 739-8521 Japan |
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| Kazushi Ueda | |
| Graduate School of Mathematical
Sciences University of Tokyo 3-8-1 Komaba Meguro-ku 153-8914 Japan |
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