A parafermionic hypergeometric function and supersymmetric 6j-symbols

Abstract

We study properties of a parafermionic generalization of the hyperbolic hypergeometric function appearing as the most important part in the fusion matrix for Liouville field theory and the Racah-Wigner symbols for the Faddeev modular double. We show that this generalized hypergeometric function is a limiting form of the rarefied elliptic hypergeometric function $V^{(r)}$ and derive its transformation properties and a mixed difference-recurrence equation satisfied by it. At the intermediate level we describe symmetries of a more general rarefied hyperbolic hypergeometric function. An important $r=2$ case corresponds to the supersymmetric hypergeometric function given by the integral appearing in the fusion matrix of $N=1$ super Liouville field theory and the Racah-Wigner symbols of the quantum algebra ${\mathrm{U}}_q(\mathrm{osp}(1|2))$. We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations.

arXiv comment29 pages

References in corpus (17)

• Liouville theory revisited
• Discrete Heisenberg-Weyl Group and Modular Group
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• Essays on the theory of elliptic hypergeometric functions
• Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
• Global Properties of Supersymmetric Theories and the Lens Space
• S^3/Z_n partition function and dualities
• Localization on three-dimensional manifolds
• Properties of generalized univariate hypergeometric functions
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• The universal Racah-Wigner symbol for Uq(osp(1|2))
• Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks
• The Endless Beta Integrals
• Parafermionic polynomials, Selberg integrals and three-point correlation function in parafermionic Liouville field theory
• General modular quantum dilogarithm and beta integrals
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group

Cited by in corpus (4)

• 3d gravity from Virasoro TQFT: Holography, wormholes and knots
• S-move matrix in the NS sector of $N=1$ super Liouville field theory
• Complex rational Ruijsenaars model. The two-particle case
• Regge symmetry of 6j-symbols of the Lorentz group