A parafermionic hypergeometric function and supersymmetric 6j-symbols

TL;DR

This paper introduces a parafermionic hypergeometric function linked to Liouville field theory and supersymmetric quantum groups, exploring its properties, symmetries, and relations to known functions and conjectures.

Contribution

It presents a new parafermionic hypergeometric function as a limit of elliptic hypergeometric functions, analyzes its properties, and connects it to supersymmetric Racah-Wigner symbols and known symmetries.

Findings

Derived transformation properties and difference equations for the new hypergeometric function.

Established relations to supersymmetric hypergeometric functions in super Liouville theory.

Proved conjectures regarding supersymmetric Racah-Wigner symbols and their parametrizations.

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 ${\rm U}_q({\rm osp}(1|2))$. We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations.