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 C(slr+1, k). In this work we classify the braided autoequivalences of the categories of local modules for all known type I quantum subgroups of C(slr+1, k), barring a single unresolved case for an orbifold of C(sl4, 8). We find that the symmetries are all non-exceptional except for four possible cases (up to level-rank duality). These exceptional cases are the orbifolds C(sl2, 16) 0Rep(Z 2 ) , C(sl3, 9) 0 Rep(Z 3 ) , C(sl4, 8) 0 Rep(Z 4 ) , and C(sl5, 5) 0Rep(Z 5 ) . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C(slr+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 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, and to reduce the C(sl4, 8) 0Rep(Z 4 ) case to a concrete finite computation. 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 C(slr+1, k). When paired with Gannon's type I classification for r ≤ 6, this will complete the type II classification for these same ranks, excluding the one exception at C(sl4, 8).

Graded extensions of generalized Haagerup categories

“…It was shown in [Izu18] that there is a unique generalized Haagerup category C for G = Z 2 × Z 2 . This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. It was shown in [Gro19] that the Brauer-Picard group of this category has order 360, and the group was identified as S 3 × A 5 in [Edi21a].…”

“…This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. It was shown in [Gro19] that the Brauer-Picard group of this category has order 360, and the group was identified as S 3 × A 5 in [Edi21a]. The outer automorphism subgroup is A 4 .…”

“…Our other application is to the generalized Haagerup category for the group Z 2 × Z 2 . This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. This category is interesting because its Brauer-Picard group is unusually rich: it was shown in [Gro19] that this group has order 360, and it was identified as S 3 × A 5 in [Edi21a].…”