Classification of ${\mathfrak{sl}}_3$ relations in the Witt group of nondegenerate braided fusion categories

Abstract

The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup ${\mathcal{W}}_{\text{un}}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}({\mathfrak{sl}}_2,k)]$, $k\ge 1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}({\mathfrak{sl}}_3,k)]$, $k\ge 1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $\mathfrak{g}$ and positive integer levels $k$.

References in corpus (1)

• Braided tensor categories and extensions of vertex operator algebras