Reconstructing Braided Subcategories of $SU(N)_k$

TL;DR

This paper extends the classification of fusion categories related to $SU(N)_k$ to include braided subcategories, providing new insights into their structure and finiteness properties.

Contribution

It generalizes Kazhdan and Wenzl's classification to braided subcategories of $SU(N)_k$, advancing understanding of their structure and classification.

Findings

Classification of braided subcategories of $SU(N)_k$

Finiteness results for these subcategories

Extension of Kazhdan-Wenzl results

Abstract

Ocneanu rigidity implies that there are finitely many (braided) fusion categories with a given set of fusion rules. While there is no method for determining all such categories up to equivalence, there are a few cases for which can. For example, Kazhdan and Wenzl described all fusion categories with fusion rules isomorphic to those of $SU(N)_k$. In this paper we extend their results to a statement about braided fusion categories, and obtain similar results for certain subcategories of $SU(N)_k$.