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Consistency conditions for dimer models. (English) Zbl 1244.14042
Let \(X\) be a three-dimensional normal Gorenstein singularity admitting a crepant resolution \(\tilde X\rightarrow X\). Then one is interested in describing the bounded derived category \(\mathcal D\text{Coh}\tilde X\) of coherent sheaves on \(\tilde X\). Bridgeland shows that this category depends only on the singularity and not on the choice of crepant resolution. In many cases there exists a tilting bundle \(\mathcal X\in\mathcal D\text{Coh}\tilde X\) such that \(\mathcal D\text{Coh}\tilde X\) is equivalent as a triangulated category to the derived category of finitely generated \(A\)-modules \(\mathcal D\text{Mod}A\), where \(A=\text{End}\mathcal X\). This leads to Van den Bergh’s introduction of a noncommutative crepant resolution (NCCR) of \(X\) which is a homologically homogeneous algebra of the form \(A=\text{End}(T)\), where \(T\) is a reflexive \(R\)-module, with \(R=\mathbb C[X]\) the coordinate ring of the singularity. However, a NCCR is not unique, and there are an infinite number of different noncommutative crepant resolutions.
To restrictions are made. The first is that \(X\) is a toric three-dimensional singularity. This implies the existence of a commutative crepant resolution. The second restriction is that the tilting bundle is a direct sum of nonisomorphic line bundles. From string theory (Franco, Hanany, Kennaway, Herzog, Vegh, Wecht), it follows that under these conditions the algebra \(A\) can be described using a dimer model on a torus. This means that \(A\) is the path algebra of a quiver \(Q\) with relations where \(Q\) is embedded in a two-dimensional torus \(T\) such that every connected piece of \(T\setminus Q\) is bounded by a cyclic path of length at least \(3\). The relations are given by demanding that for every arrow \(a\) that \(p=q\) where \(ap\) and \(aq\) are the bounding cycles that contain \(a\).
This description follows from the fact that the algebra \(A\) is a toric order, a special type of order compatible with the toric structure, and Calabi-Yau-3 (CY-3), i.e. it admits a self-dual bimodule resolution of length \(3\). It is known that every toric CY-3 order comes from a dimer model.
Under specific consistency conditions, a dimer model gives a noncommutative crepant resolution of its center. Quite different consistency conditions have been proposed. These includes cancellation, nonintersecting zig and zag rays, consistent R-charges and algebraic consistency. The aim of this article is to show that for dimer models on a torus, all these consistency conditions are equivalent. Also, the condition of being an order and the condition of being an NCCR are also equivalent to these consistency conditions. For the CY condition, the situation is less clear. The article contains an example of an infinite dimer model that is not cancellation but satisfies a suitable generalization of the CY-3 property to the infinite case. No finite counterexamples are known.
If one broadens the definition of a dimer model to allow other compact surfaces, the consistency conditions are no longer equivalent. The author discuss the differences for those cases.
The article contains a detailed introduction to quivers with relations and in particular to dimer models on a torus. Then the different consistency conditions are treated, and the known equivalences is stated and illustrated with examples. The concept of orders is treated in particular, and a main result of the article is that a Jacobi algebra of a dimer model on a torus is an order if and only if it is algebraically consistent. Finally, the author proves that the Jacobi algebra of a dimer model on a torus is a noncommutative crepant resolution of its center if and only if it is cancellation. A very precise summary of the different equivalences is given at the end. In addition to its aim of proving equivalences, the article is interesting because of its introduction to the different concepts and applications of dimer models. It triggers the reader to a further study on the subjects introduced.

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
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