Dimer models and Calabi-Yau algebras

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.