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.

29 pages

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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