Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups

TL;DR

This paper explores the relationship between global dimensions of fusion categories linked to Lie groups and quantum superfactorials, revealing new formulas and connections to conformal field theory and quantum subgroups.

Contribution

It introduces new expressions for global dimensions of fusion categories using Lie quantum superfactorials and relates them to conformal embeddings and quantum subgroups.

Findings

Global dimensions expressed via Lie quantum superfactorials.

Classical and quantum Weyl denominators linked to factorials of exponents.

Connections established between quiver periodicity and fusion rules.

Abstract

Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embeddings (they are "quantum subgroups" of a particular kind). As a side result, we express the classical (or quantum) Weyl denominator of simple Lie groups in terms of products of classical (or quantum) factorials calculated for the exponents of the group. Some calculations use the correspondence existing between periodic quivers for simply-laced Lie groups and fusion rules for module-categories (alias nimreps) of type SU(2).