Relations in the Witt Group of Nondegenerate Braided Fusion Categories Arising from the Representation Theory of Quantum Groups at Roots of Unity

Abstract

For each finite dimensional Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a modular tensor category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integrable $\hat{\mathfrak{g}}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$ in the Witt group of nondegenerate braided fusion categories have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}(\mathfrak{sl}_3,k)]$ relying on the classification of conncted \'etale alegbras in $\mathcal{C}(\mathfrak{sl}_3,k)$ ($SU(3)$ modular invariants) given by Gannon. We then give an upper bound on the levels for which exceptional connected \'etale algebras may exist in the remaining rank 2 cases ($\mathcal{C}(\mathfrak{so}_5,k)$ and $\mathcal{C}(\mathfrak{g}_2,k)$) in hopes of a future classification of Witt group relations among the classes $[\mathcal{C}(\mathfrak{so}_5,k)]$ and $[\mathcal{C}(\mathfrak{g}_2,k)]$. This dissertation contains previously published material.