Level bounds for exceptional quantum subgroups in rank two

TL;DR

This paper proves that for rank 2 Lie algebras, the modular tensor categories contain only finitely many levels with exceptional connected étale algebras, confirming a long-standing conjecture in this case.

Contribution

It establishes the finiteness of levels with exceptional subgroups for rank 2 Lie algebras, providing explicit bounds for types B2 and G2, extending previous results.

Findings

Finiteness of levels with exceptional subgroups for rank 2 cases

Explicit bounds for types B2 and G2

Extension of known results from types A1 and A2

Abstract

There is a long-standing belief that the modular tensor categories $\mathcal{C}(\mathfrak{g},k)$, for $k\in\mathbb{Z}_{\geq1}$ and finite-dimensional simple complex Lie algebras $\mathfrak{g}$, contain exceptional connected \'etale 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 $A_2$.