Faddeev–Volkov solution of the Yang–Baxter equation and discrete conformal symmetry

Bazhanov, Vladimir V.; Mangazeev, Vladimir V.; Sergeev, S. M.

doi:10.1016/j.nuclphysb.2007.05.013

Cited by 91 publications

(217 citation statements)

References 56 publications

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“…An important result in [17,18] states that the saddle point equation of the effective twisted superpotential can be interpreted as a gluing condition at the vertices of the quiver diagram of (non-ideal) hyperbolic tetrahedra whose projection to ∂H 3 is a triangle in Figure 2. The necessary and sufficient conditions for the existence of the solution of the gluing condition is stated in [18,Theorem 3].…”

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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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confidence: 99%

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Terashima

Yamazaki

2012

Phys. Rev. Lett.

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“…The R-matrix of the Faddeev-Volkov model is expressed through this function [2,20]. A number of identities for the D-function are contained in [6].…”

Section: Modular Doublementioning

confidence: 99%

“…Let us note that the symmetry restriction imposed by U q (sℓ 2 ) is not sufficient to fix completely (up to an inessential normalization factor) the R-operator. Initially the R-operator for the modular double acting on a tensor product of two infinite-dimensional representations π s (1) ⊗ π s (2) was constructed in [6] in a form similar to Eq. (4) where the role of the Euler beta function is played by the D-function, Eq.…”

Section: Trigonometric Factorizationmentioning

confidence: 99%

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Matrix Factorization for Solutions of the Yang–Baxter Equation

Derkachov

Chicherin

2016

J Math Sci

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We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finitedimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sℓ 2 , the modular double (trigonometric deformation) and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sℓ 2 , finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed out of Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulae.

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“…[20,3,4,7,19,6,5,18,17,16]. Some of those functions, particularly the one known under the name 'quantum dilogarithm' [11], also find applications in quantum topology, see e.g.…”

Section: Introductionmentioning

confidence: 99%

The Yang–Baxter relation and gauge invariance

Kashaev¹

2016

J. Phys. A: Math. Theor.

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Abstract. Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group A, we construct an operator solution of the Yang-Baxter equation generalizing the solution of the Faddeev-Volkov model. Based on a specific choice of a subgroup B ⊂ A and by using the Weil transformation, we also give a new non-operator interpretation of the Yang-Baxter relation. That allows us to construct a lattice QFT-model of IRF-type with gauge invariance under independent B-translations of local 'spin' variables.

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Faddeev–Volkov solution of the Yang–Baxter equation and discrete conformal symmetry

Cited by 91 publications

References 56 publications

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Matrix Factorization for Solutions of the Yang–Baxter Equation

The Yang–Baxter relation and gauge invariance

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