Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain

Kharchev, S.; Лебедев, Д. В.; Semenov-Tian-Shansky, Michael

doi:10.1007/s002200100592

Cited by 166 publications

(254 citation statements)

References 38 publications

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“…In the case of the Toda lattice, these conditions were obtained in [1][2][3][4], by finding appropriate solutions of the one-dimensional quantum Baxter equation associated to the integrable system. The Toda lattice has a relativistic generalization [5], and the techniques of separation of variables lead to a Baxter equation involving difference operators [6,7]. In the relativistic case, the solution to the quantum Baxter equation is not known, even for the two-particle lattice.…”

Section: Introductionmentioning

confidence: 99%

Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N = 3. Our proposal can be generalized to arbitrary toric Calabi-Yau manifolds and might solve the corresponding quantum integrable system of Goncharov and Kenyon.

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Section: Introductionmentioning

confidence: 99%

Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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“…A more complicated function is needed when q lies on the unit circle, |q| = 1 [15,16,17,18]. These functions, related to the Barnes gamma function of the second order, are actively used in the modern mathematical physics in the description of quantum integrable models and representations of quantum algebras [19,20,21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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Abstract. We give a brief review of the main results of the theory of elliptic hypergeometric functions -a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

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“…In [3], Faddeev introduced the modular double-the quantum algebra U q (sl 2 )⊗Uq−1(sl 2 ), wherẽ q is a modular transform of q (see also [6]). Originally, this construction was aimed at dealing with the well-defined logarithms of quantum algebra generators, which can be traced to the demand of the analytical uniqueness of the algebra representation modules.…”

Section: §1 Introductionmentioning

confidence: 99%

Continuous biorthogonality of an elliptic hypergeometric function

Spiridonov

2009

St. Petersburg Math. J.

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Abstract. A family of continuous biorthogonal functions related to an elliptic analog of the Gauss hypergeometric function is constructed. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. The relationship with the Sklyanin algebra and elliptic analogs of the Faddeev modular double are discussed in detail. §1. Introduction Classical and quantum completely integrable systems serve as a rich source of special functions. The complexity of these systems correlates with the complexity of solutions of the Yang-Baxter equation -rational, trigonometric, or elliptic [22]. In the theory of special functions this structural hierarchy is reflected in the existence of the plain and q-hypergeometric functions [1] and their elliptic generalizations [19]. One of our aims in the present paper is to clarify certain questions about this correspondence at the elliptic level.In [17,19], the author introduced an elliptic analog of the Gauss hypergeometric function V (t 1 , . . . , t 8 ) that generalizes many special functions of hypergeometric type obeying "classical" properties [1]. In particular, it was shown that this function exhibits symmetry transformations tied to the exceptional root system E 7 and satisfies the elliptic hypergeometric equation. We start this work with showing in §2 that the V -function satisfies a simple biorthogonality relation corresponding to the continuous values of a spectral parameter. For that we use the elliptic beta integral [16] and an integral analog of the Bailey chains introduced in [18]. Our construction can be viewed as an integral generalization of Rosengren's approach to the elliptic 6j-symbols [11].In §3, we investigate the generalized eigenvalue problem for a second order finite difference operator D (a, b, c, d; p; q). The relationship of such problems with biorthogonality relations was considered in [23]. A problem similar to ours was investigated earlier in [10,11] for the discrete spectrum. Its general solution determines a basis in some space of meromorphic functions. For a simple scalar product, we derive the dual basis defined by solutions of a similar spectral problem for conjugate operators. As a result, the overlap of the basis vectors with their duals coincides with the V -function. In §4, we derive a new contiguous relation for the elliptic hypergeometric function and the biorthogonality condition associated with that.It turns out that the V -function is directly related to the Sklyanin algebra, one of the central objects in the quantum inverse scattering method [14,15]. This relationship is established in §5 on the basis of the observation, due to Rains [9,12], that the D-operator can be represented as a linear combination of all four generators of the Sklyanin algebra.2000 Mathematics Subject Classification. Primary 33C75, 81R12.

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Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain

Cited by 166 publications

References 38 publications

Exact quantization conditions for the relativistic Toda lattice

Exact quantization conditions for the relativistic Toda lattice

Essays on the theory of elliptic hypergeometric functions

Continuous biorthogonality of an elliptic hypergeometric function

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