Unit Circle Elliptic Beta Integrals

Diejen, J. F. van; Spiridonov, V. P.

doi:10.1007/s11139-005-4846-x

Cited by 33 publications

(58 citation statements)

References 24 publications

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“…It is easy to check [38], that the function G(u; ω) = e − πi 3 B3,3(u|ω) Γ(e −2πi u ω 3 ;r,p), (2.21) where |p|, |r| < 1, satisfies the same three equations and the normalization as the function (2.20). Therefore these functions coincide, and their equality constitutes one of the laws of the modular transformations for the elliptic gamma function related to the SL(3; Z)-group [25].…”

Section: The Elliptic Gamma Functionsmentioning

confidence: 98%

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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Section: The Elliptic Gamma Functionsmentioning

confidence: 98%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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“…(1.1). The partition function can then be rewritten in terms of a modified elliptic hypergeometric integral introduced in [43].…”

Section: Jhep07(2017)041mentioning

confidence: 99%

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Brünner

Regalado

Spiridonov

2017

J. High Energ. Phys.

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We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3, Z) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N = 2 SCFT with E 6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N = 1 theory with enhanced E 7 flavour symmetry.

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“…In a follow-up paper [5], van Diejen and Spiridonov proved "hyperbolic" degenerations of these integrals (after the hyperbolic gamma function of Ruijsensaars [18], and results of Stokman [22] at the univariate level). Unfortunately, the asymptotic estimates they had available were insufficient to derive these hyperbolic integrals as limits of the elliptic integrals; instead, they were forced to degenerate the elliptic proofs.…”

Section: Introductionmentioning

confidence: 98%

Limits of elliptic hypergeometric integrals

Rains

2007

Ramanujan J

219

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In [16], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.

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Unit Circle Elliptic Beta Integrals

Abstract: Abstract. We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.

Cited by 33 publications

References 24 publications

Essays on the theory of elliptic hypergeometric functions

Essays on the theory of elliptic hypergeometric functions

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Limits of elliptic hypergeometric integrals

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