Prime decomposition of modular tensor categories of local modules of type D

Abstract

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k{)}_R^0$ where $R$ is the regular algebra of Tannakian Rep$(H)\subset \mathcal{C}(\mathfrak{g},k{)}_{\text{pt}}$. We describe the decomposition of $\mathcal{C}(\mathfrak{g},k{)}_R^0$ into prime factors, and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $\mathcal{C}({\mathfrak{so}}_5,k)$ and $\mathcal{C}({\mathfrak{g}}_2,k)$ for $k\in {\mathbb{Z}}_{\ge 1}$.