Dimer models and Calabi-Yau algebras

Abstract

In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. We prove that the algebra obtained from an algebraically consistent dimer model is a 3-dimensional Calabi-Yau algebra and finally prove that this gives a non-commutative crepant resolution of the Gorenstein affine toric threefold associated to the dimer model.

93 pages - Version to appear in Memoirs of the AMS (accepted 5th October 2009). Chapter 8 added including proof that every Gorenstein affine toric threefold admits a non-commutative crepant resolution which can be obtained from a dimer model. Sections 1.3 and 3.4.1 added. Chapters 4 and 5 exchanged. Figures added

References in corpus (4)

• Brane Tilings
• Properly ordered dimers, $R$-charges, and an efficient inverse algorithm
• Computation of Principal A-determinants through Dimer Dynamics
• Noncommutative resolutions and rational singularities

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• Crepant resolutions and brane tilings II: Tilting bundles
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