The Yang–Baxter relation and gauge invariance

Kashaev, Rinat

doi:10.1088/1751-8113/49/16/164001

Cited by 15 publications

(20 citation statements)

References 24 publications

(37 reference statements)

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“…In a series of papers [2][3][4]15], topological invariants of (ideally triangulated) 3-manifolds have been constructed from certain self-dual LCA groups equipped with quantum dilogarithm functions. The main idea of those constructions is the following.…”

Section: Discussionmentioning

confidence: 99%

“…In this subsection, we identify the tetrahedral weight function ψ 0 (z, w) with (the reciprocal of) the quantum dilogarithm function ψ(z, m) on the self-dual LCA group T × Z, given by [15,Eqn.97…”

Section: The Quantum Dilogarithmmentioning

confidence: 99%

“…RHS is absolutely convergent in the domain (17) and it can be explicitly calculated by using Ramanujan's 1 ψ 1 -summation formula. The detailed computation appears in [15,Eqn. (98)], and the result reads…”

Section: Theorem 25 (A)mentioning

confidence: 99%

“…Appendix A: A quantum dilogarithm over the LCA group T × Z The function (19) with q real has been introduced and studied in the functional analytic context of Hilbert spaces and quantum E(2) group by Woronowicz in [21]. In this section, we derive some of its operator properties by using the theory of quantum dilogarithms over Pontryagin self-dual LCA groups developed in [2,15].…”

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confidence: 99%

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A meromorphic extension of the 3D index

Garoufalidis

Kashaev

2018

Res Math Sci

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Using the locally compact abelian group T × Z, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2-3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a strict angle structure, our meromorphic function can be expanded into a Laurent power series whose coefficients are formal power series in q with integer coefficients that coincide with the 3D index of (Dimofte et al. in Adv Theor Math Phys 17(5):975-1076, 2013). Our meromorphic function can be computed explicitly from the matrix of the gluing equations of a triangulation, and we illustrate this with several examples.

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Section: Discussionmentioning

confidence: 99%

Section: The Quantum Dilogarithmmentioning

confidence: 99%

Section: Theorem 25 (A)mentioning

confidence: 99%

mentioning

confidence: 99%

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A meromorphic extension of the 3D index

Garoufalidis

Kashaev

2018

Res Math Sci

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“…Before going into details of this correspondence it is useful to refer to other recent remarkable appearances of the star-triangle relation (and Yang-Baxter equation, in general) in different and seemingly unrelated areas of physics and mathematics. In particular, there are deep connections to the theory of elliptic hypergeometric functions [16][17][18], topological quantum field theory [19][20][21] and calculations of superconformal indices connected with electric-magnetic dualities in 4D N = 1 superconformal Yang-Mills theories [22]. Indeed, as found in [23][24][25][26], the 4D superconformal quiver gauge theories are closely related to previously known 2D lattice models [12,27] and also lead to rather non-trivial new ones [28][29][30][31].…”

Section: Introductionmentioning

confidence: 97%

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Bazhanov¹,

Kels²,

Сергеев³

2016

J. Phys. A: Math. Theor.

109

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In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values on the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the startriangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.

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The star-triangle relation, lens partition function, and hypergeometric sum/integrals

Gahramanov

Kels

2017

J. High Energ. Phys.

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The aim of the present paper is to consider the hyperbolic limit of an elliptic hypergeometric sum/integral identity, and associated lattice model of statistical mechanics previously obtained by the second author. The hyperbolic sum/integral identity obtained from this limit, has two important physical applications in the context of the so-called gauge/YBE correspondence. For statistical mechanics, this identity is equivalent to a new solution of the star-triangle relation form of the Yang-Baxter equation, that directly generalises the Faddeev-Volkov models to the case of discrete and continuous spin variables. On the gauge theory side, this identity represents the duality of lens (S 3 b /Z r ) partition functions, for certain three-dimensional N = 2 supersymmetric gauge theories.

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The Yang–Baxter relation and gauge invariance

Cited by 15 publications

References 24 publications

A meromorphic extension of the 3D index

A meromorphic extension of the 3D index

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

The star-triangle relation, lens partition function, and hypergeometric sum/integrals

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