Dimer models and Calabi-Yau algebras

Broomhead, Nathan

doi:10.1090/s0065-9266-2011-00617-9

Cited by 116 publications

(235 citation statements)

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“…In [5], a distinction is made between geometrically consistent and marginally consistent R-charges. The former have the extra condition that R a < 1 for every a, while for the latter, one also allows R a ≥ 1.…”

Section: A0mentioning

confidence: 99%

“…REMARK 7.8. The idea of cutting out a piece bounded by paths that do not meet a certain perfect matching is borrowed from Section 6.3.1 in [5]. In order to make this work in the marginal consistent case, we used the new notion of these P θ which do not appear in [5].…”

Section: Is Algebraically Consistent If and Only If It Is Cancellationmentioning

confidence: 99%

“…several quite different consistency conditions were proposed such as cancellation [7], nonintersecting zig and zag rays [16], consistent R-charges [15] and algebraic consistency [5].…”

mentioning

confidence: 99%

“…Moreover, the condition of being an order and the condition of being an NCCR are also equivalent to these consistency conditions. The situation for the Calabi-Yau condition is less clear: it was shown by Davison [7] and Broomhead [5] that cancellation and algebraic consistency imply the CY-3 condition, but there is no proof for the other direction. We will give 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.…”

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confidence: 99%

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Consistency Conditions for Dimer Models

Bocklandt

2012

Glasgow Math. J.

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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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Section: A0mentioning

confidence: 99%

Section: Is Algebraically Consistent If and Only If It Is Cancellationmentioning

confidence: 99%

“…several quite different consistency conditions were proposed such as cancellation [7], nonintersecting zig and zag rays [16], consistent R-charges [15] and algebraic consistency [5].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Consistency Conditions for Dimer Models

Bocklandt

2012

Glasgow Math. J.

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“…In fact, Broomhead [Bro09] constructs a non-commutative crepant resolution for every Gorenstein affine toric threefold, from superpotential algebras called dimer models. Similar algebras associated to brane tilings have non-commutative crepant resolutions as well [Moz09,BM09].…”

Section: S Omissions and Open Questionsmentioning

confidence: 99%

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Leuschke

2012

Progress in Commutative Algebra 1

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ABSTRACT. Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.

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Equivalence of a-maximization and volume minimization

Eager

2014

J. High Energ. Phys.

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The low energy effective theory on a stack of D3-branes at a Calabi-Yau singularity is an N = 1 quiver gauge theory. The AdS/CFT correspondence predicts that the strong coupling dynamics of the gauge theory is described by weakly coupled type IIB supergravity on AdS 5 × L 5 , where L 5 is a Sasaki-Einstein manifold. Recent results on Calabi-Yau algebras efficiently determine the Hilbert series of any superconformal quiver gauge theory. We use the Hilbert series to determine the volume of the horizon manifold in terms of the fields of the quiver gauge theory. One corollary of the AdS/CFT conjecture is that the volume of the horizon manifold L 5 is inversely proportional to the a-central charge of the gauge theory. By direct comparison of the volume determined from the Hilbert series and the a-central charge, this prediction is proved independently of the AdS/CFT conjecture.

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Dimer models and Calabi-Yau algebras

Cited by 116 publications

References 33 publications

Consistency Conditions for Dimer Models

Consistency Conditions for Dimer Models

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Equivalence of a-maximization and volume minimization

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