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

56 pages, v2: major revision

References in corpus (17)

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• Calabi-Yau algebras
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• McKay correspondence
• Consistency conditions for brane tilings
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Cited by in corpus (33)

• Derived categories of small toric Calabi-Yau 3-folds and counting invariants
• Crepant resolutions and brane tilings I: Toric realization
• A Dimer ABC
• A note on consistency conditions on dimer models
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Dimer models and exceptional collections
• Crepant resolutions and brane tilings II: Tilting bundles
• Dimers and cluster integrable systems
• Wall-crossing of the motivic Donaldson-Thomas invariants
• Geometric engineering of (framed) BPS states
• Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions
• Remarks on quiver gauge theories from open topological string theory
• Non-commutative Donaldson-Thomas theory and vertex operators
• Combinatorial Reid's recipe for consistent dimer models
• Cancellativization of dimer models
• Deformations of Dimer Models
• Dimer algebras, ghor algebras, and cyclic contractions
• On steady non-commutative crepant resolutions
• New dimer integrable systems and defects in five dimensional gauge theory
• Mutations of splitting maximal modifying modules: The case of reflexive polygons
• Exact Lefschetz fibrations associated with dimer models
• Nonlocality and the central geometry of dimer algebras
• Walls for $G$-Hilb via Reid's Recipe
• Dimer models and crepant resolutions
• Remarks on divisorial ideals arising from dimer models
• Geometric Reid's recipe for dimer models
• Consistent Dimer Models on Surfaces with Boundary
• Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb')
• Strebel Differentials and stable Matrix Factorizations
• Non-commutative crepant resolutions of Hibi rings with small class group
• Cyclic contractions of dimer algebras always exist
• Semi-steady non-commutative crepant resolutions via regular dimer models
• Gale duality and the linearisation map for noncommutative crepant resolutions