Level bounds for exceptional quantum subgroups in rank two

Abstract

There is a long-standing belief that the modular tensor categories $\mathcal{C}(\mathfrak{g},k)$, for $k\in {\mathbb{Z}}_{\ge 1}$ and finite-dimensional simple complex Lie algebras $\mathfrak{g}$, contain exceptional connected étale algebras at only finitely many levels $k$. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here we confirm this conjecture when $\mathfrak{g}$ has rank 2, contributing proofs and explicit bounds when $\mathfrak{g}$ is of type $B_2$ or $G_2$, adding to the previously known positive results for types $A_1$ and $A_2$.

References in corpus (5)

• Braided tensor categories and extensions of vertex operator algebras
• Consistent systems of correlators in non-semisimple conformal field theory
• Quantum symmetries for exceptional SU(4) modular invariants associated with conformal embeddings
• Exceptional quantum subgroups for the rank two Lie algebras B2 and G2
• Classification of ${\mathfrak{sl}}_3$ relations in the Witt group of nondegenerate braided fusion categories

Cited by in corpus (3)

• Higher Gauss sums of modular categories
• Subfactors and Mathematical Physics
• Type $II$ quantum subgroups of ${\mathfrak{sl}}_N$. $I$: Symmetries of local modules