Auto-equivalences of the modular tensor categories of type $A$, $B$, $C$ and $G$

Abstract

We compute the monoidal and braided auto-equivalences of the modular tensor categories $\mathcal{C}({\mathfrak{sl}}_{r+1},k)$, $\mathcal{C}({\mathfrak{so}}_{2r+1},k)$, $\mathcal{C}({\mathfrak{sp}}_{2r},k)$, and $\mathcal{C}({\mathfrak{g}}_2,k)$. Along with the expected simple current auto-equivalences, we show the existence of the charge conjugation auto-equivalence of $\mathcal{C}({\mathfrak{sl}}_{r+1},k)$, and exceptional auto-equivalences of $\mathcal{C}({\mathfrak{so}}_{2r+1},2)$, $\mathcal{C}({\mathfrak{sp}}_{2r},r)$, $\mathcal{C}({\mathfrak{g}}_2,4)$. We end the paper with a section discussing potential applications of these computations.

51 pages, a new appendix by Terry Gannon has been added, comments welcome