Brane tilings and reflexive polygons

Hanany, Amihay; Seong, Rak-Kyeong

doi:10.1002/prop.201200008

Cited by 70 publications

(113 citation statements)

References 80 publications

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“…In this subsection we verify that the toric quiver gauge theories constructed in [21,26] are indeed a subclass of solutions of the Diophantine equation presented here. These models, being toric, have the same rank N in all blocks.…”

Section: Reproducing Known Theoriessupporting

confidence: 57%

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Superconformal block quivers, duality trees and Diophantine equations

Hanany

Sun

et al. 2013

J. High Energ. Phys.

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We generalize previous results on N = 1, (3 + 1)-dimensional superconformal block quiver gauge theories. It is known that the necessary conditions for a theory to be superconformal, i.e. that the beta and gamma functions vanish in addition to anomaly cancellation, translate to a Diophantine equation in terms of the quiver data. We re-derive results for low block numbers revealing an new intriguing algebraic structure underlying a class of possible superconformal fixed points of such theories. After explicitly computing the five block case Diophantine equation, we use this structure to reorganize the result in a form that can be applied to arbitrary block numbers. We argue that these theories can be thought of as vectors in the root system of the corresponding quiver and superconformality conditions are shown to associate them to certain subsets of imaginary roots. These methods also allow for an interpretation of Seiberg duality as the action of the affine Weyl group on the root lattice. * a.hanany@imperial.ac.uk † hey@maths.ox.ac.uk

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Section: Reproducing Known Theoriessupporting

confidence: 57%

“…Note though that the quivers studied here are more general since they are not necessarily Calabi-Yau threefolds. In the toric subclass of Calabi-Yau manifolds, due to the combinatorial nature of the geometry, attempts are under way towards an enumeration [21,[25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Superconformal block quivers, duality trees and Diophantine equations

Hanany

Sun

et al. 2013

J. High Energ. Phys.

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“…On the other side, by [28,Theorem 3.12], one can reconstruct Newton polygon from dimer partition function of the bipartite graph. This correspondence between polygons with single interior point, bipartite graphs, and quivers is not new, see [23,32] and references therein.…”

Section: Polygons With Single Interior Pointmentioning

confidence: 98%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

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“…3 Zv 3 σ 0 = (1, 2, 3)(4, 5, 6) (7,8,9) (10,11,12) σ 1 = (1, 11, 9)(4, 8, 12)(7, 2, 6)(10, 5, 3) This is a lattice in R 2 if F is convex. For every sublattice Λ ⊂ Aut(F) the function F descends to a function on the torus R 2 /Λ and gives a tiling of this torus by black and white polygons.…”

Section: Zhegalkin Zebra Functionsmentioning

confidence: 99%

Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry

Stienstra¹

2018

SIGMA

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Zhegalkin zebra motives are tilings of the plane by black and white polygons representing certain F 2 -valued functions on R 2 . They exhibit a rich geometric structure and provide easy to draw insightful visualizations of many topics in the physics and mathematics literature. The present paper gives some pieces of a general theory and a few explicit examples. Many more examples will be shown in the forthcoming article "Zhegalkin zebra motives: algebra and geometry in black and white".

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Brane tilings and reflexive polygons

Cited by 70 publications

References 80 publications

Superconformal block quivers, duality trees and Diophantine equations

Superconformal block quivers, duality trees and Diophantine equations

Cluster integrable systems, q-Painlevé equations and their quantization

Zhegalkin Zebra Motives Digital Recordings of Mirror Symmetry

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