Dimer models and Calabi-Yau algebras

Broomhead, Nathan

doi:10.48550/arxiv.0901.4662

Cited by 35 publications

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“…2 Moreover, the module category corresponds to the category of perverse sheaves under this derived equivalence. This is stronger than what we can get from the general results such as [MR10,Dav,Bro,IUa].…”

Section: Introductionmentioning

confidence: 58%

DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

Nagao

2012

The Quarterly Journal of Mathematics

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We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent sheaves. As an application we provide some equations on Donaldson-Thomas, Pandharipande-Thomas and Szendroi's invariants. 3 Kontsevich-Soibelman's (partly conjectural) formula ([KS]) also covers the setting in this paper. 4 After this paper submitted • Euler characteristic version of the DT-PT conjecture was proved by Toda, Thomas-Stoppa, Bridgeland ([Tod10, ST, Bri]) using Joyce's arguments, and • Joyce-Song provided an extension of the wall-crossing formula in [Joy08] to weighted Euler characteristics and an application for non-commutative Donaldson-Thomas invariants ([JS]).

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Section: Introductionmentioning

confidence: 58%

DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

Nagao

2012

The Quarterly Journal of Mathematics

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“…The main step in proving Theorem 1 is to establish a certain Calabi-Yau property for the algebra A D , relative to the boundary idempotent e, as referred to in the title. This result is likely to be of independent interest, and is analogous to Broomhead's theorem [3,Thm. 7.1] concerning Calabi-Yau properties of algebraically consistent dimer models on the torus.…”

Section: Introductionmentioning

confidence: 52%

“…Proof. Exactness of (3.1) and of (3.2) are equivalent because of the existence of a grading as in Proposition 3.1, by an argument that is essentially due to Broomhead [3,Prop. 7.5].…”

Section: The Calabi-yau Propertymentioning

confidence: 96%

“…In the case that Σ is a disc, these strands will always satisfy conditions (P0)-(P3) from Definition 2.1 (interpreting the self-intersection condition in (P3) appropriately for closed curves). Requiring that they also satisfy (P4)-which in particular rules out any closed curves in the interior-places an extra condition on the dimer model G, the analogue of Broomhead's geometric consistency [3,Prop. 3.12].…”

Section: Postnikov Diagrams and Their Dimer Algebrasmentioning

confidence: 99%

“…Thinness of A D , as in Proposition 2.8, is the analogue of algebraic consistency for dimer models on the torus, as described by Broomhead [3] (the analogue of geometric consistency, cf. [3, Prop.…”

Section: Postnikov Diagrams and Their Dimer Algebrasmentioning

confidence: 99%

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Calabi-Yau properties of Postnikov diagrams

Pressland¹

2019

Preprint

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We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally 3-Calabi-Yau in the sense of the author's earlier work [31]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian, by a recent result of Galashin and Lam [13].

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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 35 publications

References 0 publications

DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

Calabi-Yau properties of Postnikov diagrams

Equivalence of a-maximization and volume minimization

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