Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Abstract

This paper is the first of a pair that aims to classify a large number of the type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. In this work we classify the braided auto-equivalences of the categories of local modules for all known type $I$ quantum subgroups of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds $\mathcal{C}( \mathfrak{sl}_{2},16)_{\operatorname{Rep}(\mathbb{Z}_2)}$, $\mathcal{C}( \mathfrak{sl}_{3},9)_{\operatorname{Rep}(\mathbb{Z}_3)}$, $\mathcal{C}( \mathfrak{sl}_{4},8)_{\operatorname{Rep}(\mathbb{Z}_4)}$, and $\mathcal{C}( \mathfrak{sl}_{5},5)_{\operatorname{Rep}(\mathbb{Z}_5)}$. We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type $D$-$D$ case, which is used to construct one of the exceptionals. Finally we uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. When paired with Gannon's type $I$ classification for $r\leq 6$, this will complete the type $II$ classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$.