Quivers, YBE and 3-manifolds

Yamazaki, Masahito

doi:10.1007/jhep05(2012)147

Cited by 72 publications

(127 citation statements)

References 135 publications

(256 reference statements)

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“…The superconformal index of quivers for toric Calabi-Yau cones were also studied in [48,49] from a rather different perspective. It will be of interest to see what connections, if any, there are with our work.…”

Section: Discussionmentioning

confidence: 99%

Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

Eager

Schmude

Tachikawa

2014

Advances in Theoretical and Mathematical Physics

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The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.

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Section: Discussionmentioning

confidence: 99%

Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

Eager

Schmude

Tachikawa

2014

Advances in Theoretical and Mathematical Physics

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“…When the circle radii are all of the same value, the Legendre transform of the volume of M R with respect to the angles θ * e is related with the prepotential of the topological string theory on the dual toric Calabi-Yau 3-fold [4]. The correspondence between 2d twisted superpotential and the 3d classical hyperbolic geometry motivates us to propose a quantum version of the correspondence: the 3d partition function Z 3d with finite b compute the partition function of the 3d SL(2) Chern-Simons theory, where b is related to the level t by t ∼ 1/b 2 .…”

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confidence: 99%

“…Further details will be presented in a separate publication [4]. 4d versus 2d.-We begin with a 4d N = 1 quiver superconformal field theory obtained by probing toric Calabi-Yau 3-fold by N D3-branes.…”

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confidence: 99%

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Terashima

Yamazaki

2012

Phys. Rev. Lett.

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We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose an equality between (1) the 4d superconformal index of a 4d N = 1 superconformal quiver gauge theory described by a bipartite graph on T 2 and (2) the partition function of a classical integrable spin chain on T 2 . The 2d spin system is lifted to a hyperbolic 3-manifold after the dimensional reduction and the Higgsing of the 4d gauge theory.Introduction.-The concept of spacetime has been of crucial importance in our understanding of Nature. However, in the theory of quantum gravity, it is widely believed that even the notion of classical spacetime is of secondary nature, and emerges from a more fundamental structure. One proposal for such a structure is the spin network [1], a spin system defined on a discrete lattice.In a different line of development, more recently there have been important developments in supersymmetric gauge theories suggesting that the spacetime geometry could be traded for another "internal" geometry. This has been discussed for a class of supersymmetric gauge theories compactified on a compact curved manifold, which is thought of as the Euclidean version of the spacetime for the theory. The idea is simple; we begin with a D-dimensional field theory and compactify the theory on a class of d 1 -dimensional manifolds C. The resulting d 2 -dimensional theory is defined on a fixed d 2 -dimensional compact manifold S, where d 1 + d 2 = D. We could instead first compactify on S, and then we have a d 1 -dimensional theory on C. Thus we have a correspondence between the d 2 -dimensional field theory on S and the d 1 -dimensional field theory on C.While the idea itself is rather general, in practice it is a rather difficult problem to make a precise identification between the observables of the two theories, since a quantity on one side could take a rather different form on the other. A successful example of such a quantitative identification is the relation between the S 4 partition function of 4d N = 2 superconformal field theories (SCFT) arising from a compactification of 6d (2, 0) theory on a Riemann surface C [2] and a correlation function of 2d Liouville theory on C [3].The goal of this Letter is to unify these two apparently unrelated ideas in supersymmetric gauge theories and gravity. This gives new perspectives on the emergence of classical geometry, and surprisingly the process has a counterpart in the supersymmetric gauge theory.We analyze the 4d superconformal index for quiver gauge theories dual to toric Calabi-Yau 3-folds, and find that the 4d index is equivalent to the partition function of an integrable spin system in 2d. We then discuss dimensional reduction from the 4d index to the 3d partition

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“…In another direction, the lens elliptic gamma function, which is a central function for this paper, first appeared in the study of four-dimensional N = 1 supersymmetric gauge theories on a circle times the lens space S 3 /Z r [7]. This connection suggests that there exists an interpretation of the results of this paper in terms of supersymmetric gauge theories and associated integrable lattice models [8,[11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 72%

“…Elliptic Hypergeometric Sum/Integral. A central role in this paper is played by the following sum/integral, defined in terms of the lens elliptic gamma function (14), by…”

Section: 1mentioning

confidence: 99%

Lens Generalisation of τ-Functions for the Elliptic Discrete Painlevé Equation

Kels

Yamazaki

2019

International Mathematics Research Notices

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We propose a new bilinear Hirota equation for τ-functions associated with the E 8 root lattice, that provides a "lens" generalisation of the τ-functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer r in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the E 8 root lattice. We construct explicit W(E 7 )-invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals. 14 5.2. Decomposition of τ-function 16 6. Hypergeometric τ-Function 18 6.

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Quivers, YBE and 3-manifolds

Cited by 72 publications

References 135 publications

Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Lens Generalisation of τ-Functions for the Elliptic Discrete Painlevé Equation

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