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

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.