Dimer models and Calabi-Yau algebras

Details Dimer models and Calabi-Yau algebras Dimer models and Calabi-Yau algebras

2009-01-29

arxiv.org/abs/0901.4662v2

Mathematics

Algebraic Geometry

93 pages - Version to appear in Memoirs of the AMS (accepted 5th October 2009). Chapter 8 added including proof that every Gor

Scientific paper

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.

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