Dimer Models and Calabi-Yau Algebras

By Nathan Broomhead. 2009

166 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e42dc991ee2a8ffea34a99f8977329148ca088b3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Non-Commutative Crepant Resolutions: Scenes from Categorical Geometry (2011, background): “In fact, Broomhead [Bro09] constructs a non-commutative crepant resolution for every Gorenstein affine toric threefold, from superpotential algebras called dimer models.”
Non-commutative resolutions of quotient singularities for reductive groups (2015, unknown): “On the other hand if R is Gorenstein and dimSpecR = 3 then a NCCR always exists [5].”
Dimer models and cluster categories of Grassmannians (2013, methodology): “We will make this more precise in the next section, adapting the more quiver focussed formalism of Bocklandt [1, 2], Davison [7] and Broomhead [3] to the boundary case.”
A geometric model for the derived category of gentle algebras (2018, unknown): “lgebras ﬁrst appeared in the form of iterated tilted algebras of type A[6, 7] and type A˜ [8]. It has transpired since that they naturally appear in many diﬀerent contexts; these include dimer models [21, 22], enveloping algebras of Lie algebras [49], and cluster theory, where they appear as m-clust”
Dimer models and the special McKay correspondence (2009, result): “ call the second consistency condition. The latter condition is shown to be redundant by Davison [Dav11]. Broomhead has proved the Calabi-Yau 3 property of the path algebra for isoradial dimer models [Bro12]. The proof of Theorem 1.4 in this paper does not rely on any of these results, and gives an ”
A Geometric Model for the Module Category of a Gentle Algebra (2018, background): “. Gentle algebras are Gorenstein [24], closed under tilting and derived equivalence [38, 39], and they are ubiquitous, for instance they occur in contexts such as Fukaya categories [27], dimer models [8], enveloping algebras of Lie algebras [29] and cluster theory. As such, there has been widespread”