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.
Cite
@article{arxiv.0901.4662,
title = {Dimer models and Calabi-Yau algebras},
author = {Nathan Broomhead},
journal= {arXiv preprint arXiv:0901.4662},
year = {2010}
}
Comments
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