Consistency Conditions for Dimer Models

Abstract

Abstract. Dimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper, we study the major different notions in detail and show that for dimer models on a torus, they are all equivalent.2010 Mathematics Subject Classification. 14M25, 14A22, 16S38. Introduction.If X is a three-dimensional normal Gorenstein singularity admitting a crepant resolutionX → X, then one is interested to describe the bounded derived category DCohX of coherent sheaves onX. A well-known result by Bridgeland [4] shows that this category only depends on the singularity and not on the particular choice of crepant resolution.In many cases, there exists a tilting bundle in U ∈ DCohX such that DCohX is equivalent as a triangulated category to the derived category of finitely generated A-modules DModA, where A = EndX . To model these algebras without referring to a commutative crepant resolution, Van den Bergh [23] introduced the notion of a noncommutative crepant resolution (NCCR) of X. This is a homologically homogeneous algebra of the form A = End R (T), where T is a reflexive R-module, with R = ‫[ރ‬X] the coordinate ring of the singularity. An NCCR is, however, far from unique and in general there are an infinite number of different noncommutative crepant resolutions.If we make two restrictions, the problem becomes more manageable. First, we assume that X is a toric three-dimensional Gorenstein singularity. This automaticly implies the existence of a commutative crepant resolution. Secondly, we assume that the tilting bundle is a direct sum of nonisomorphic line bundles. It was noticed in string theory [8,10,11] 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 real torus T such that every connected piece of T \ Q is bounded by a cyclic path of length at least 3. The relations are given by demanding for every arrow a that p = q where ap and aq are the bounding cycles that contain a.This nice 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. In [2], it was shown that every toric CY-3 order comes from a dimer model. Not every dimer model gives a noncommutative crepant resolution of its centre. To do so, it needs to satisfy some extra conditions called consistency conditions. In recent years,

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“…The analogue problem for brane tilings and, more generally, bipartite graphs on Riemann surfaces has been extensively studied (see e.g. [35][36][37][38][39][40][41][42] and references therein). In this section we take the first steps on this issue for brane brick models, proposing natural generalizations of the brane tiling case.…”

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“…To characterize such dimer models we introduce a notion of consistency. Several different notions are available in the literature [22,12,13,36,25,11] but we will restrict to one: zigzag consistency. Fix a dimer model Q and its universal coverQ u , which is again a dimer.…”

“…In [13] Davison proved that the cancellation property alone implies the Calabi-Yau property and that this second technical condition follows from the cancellation property. In [11] it is shown that zigzag consistency implies the cancellation property. Similar results are also obtained by Ishii and Ueda in [24] and [25].…”

“…Geometric consistency can be relaxed to the notion of Rcharge consistency: the existence of a R >0 -grading R such that for every cycle c ∈ Q 2 a∈c R a = 2 and every vertex v h(a)=v (1 − R a ) + t(a)=v (1 − R a ) = 0. In [11] it is shown that R-charge consistency is equivalent to zigzag consistency on a torus.…”

“…Note that not every dimer has a perfect matching [12] but on a torus zigzag consistency implies the existence of a perfect matching [11]. If Q has a perfect matching P then ℓ is a homogeneous element of degree deg P ℓ = 2.…”