Crepant resolutions and brane tilings I: Toric realization

Abstract

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of all its commutative crepant resolutions. We also explain how the McKay correspondence in dimension 3 can be interpreted using brane tilings.

22 pages

References in corpus (8)

• Calabi-Yau algebras
• Consistency conditions for brane tilings
• Dimer models and the special McKay correspondence
• Dimer models and Calabi-Yau algebras
• On moduli spaces of quiver representations associated with dimer models
• On the noncommutative Donaldson-Thomas invariants arising from brane tilings
• D-Branes on Toric Calabi-Yau Varieties
• Resolutions of Orbifold Singularities and Flows on the McKay Quiver

Cited by in corpus (15)

• Motivic Donaldson-Thomas invariants and McKay correspondence
• A Dimer ABC
• Cellular resolutions of noncommutative toric algebras from superpotentials
• Dimer models and exceptional collections
• Crepant resolutions and brane tilings II: Tilting bundles
• On the noncommutative geometry of square superpotential algebras
• 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
• Exact Lefschetz fibrations associated with dimer models
• Attractor invariants, brane tilings and crystals
• Consistent Dimer Models on Surfaces with Boundary
• Noncommutative mirror symmetry for punctured surfaces
• Geometric Reid's recipe for dimer models
• Strebel Differentials and stable Matrix Factorizations