On the unitarity and modularity of ribbon tensor categories associated with affine Lie algebras

TL;DR

This work classifies when ribbon tensor categories derived from quantum groups of simple affine Lie algebras are (pseudo-)unitary and modular, using numerical calculations and two guiding conjectures. It distinguishes uniform and non-uniform cases by the root-length ratio and root-of-ununity parameters, computing quantum dimensions and a pseudo-modularity criterion to determine modularity. The results largely confirm known rigorous findings and fill gaps, offering a practical catalog of which $q$ (via $q=e^{2\,pi i p/\ell}$) yield UMTCs in these families. The study advances the understanding of which affine Lie–derived RTCs can underpin fault-tolerant topological quantum computation and related modular data constructions.

Abstract

We study the unitarity and modularity of ribbon tensor categories derived from simple affine Lie algebras, via their associated quantum groups. Based on numerical calculations, and assuming two conjectures, we provide the complete picture for which values of $q$ these ribbon tensor categories are (pseudo-)unitary and for which values of $q$ they are modular. We compare our results with the extensive rigorous results appearing in the literature, finding complete agreement. For the cases that do not appear in the literature, we complete the picture.