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 C(slr+1, k), C(so2r+1, k), C(sp 2r , k), and C(g2, k). Along with the expected simple current auto-equivalences, we show the existence of the charge conjugation auto-equivalence of C(slr+1, k), and exceptional auto-equivalences of C(so2r+1, 2), C(sp 2r , r), C(g2, 4). We end the paper with a section discussing potential applications of these computations.

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

“…The planar algebra P Λ 1 +Λr is well understood [36,13]. It is generated by two trivalent vertices satisfying the Thurston relations (see [36,Lemma 3.2]).…”

“…The coefficients c 1 , c 2 , c 3 , c 4 for which φ preserve the Thurston relations are solved for in [13,Lemma 3.1]. With the condition that φ is braided, there are two solutions, which we denote φ id and φ cc .…”

“…To construct the remaining auto-equivalences, we observe that the charge-conjugation autoequivalences exist for C(sl r+1 , k) ad when k ≥ 3 and r ≥ 2 [13]. These auto-equivalences preserve the Rep(Z m ′ ) subcategory, and hence descend to auto-equivalences of C(sl r+1 , k) ad Rep(Z m ′ ) .…”

“…We will construct these auto-equivalences as simple current auto-equivalences. For the definition of simple current auto-equivalences we use in this paper, see [13,Lemma 2.4].…”

“…For the four truly exceptional examples found by Gannon we can quickly compute that the auto-equivalence groups are all trivial, except for the Galois conjugate of C(sl 2 , 10) which has auto-equivalence group Z 2 [12,Theorem 1.2]. For the type I quantum subgroups coming from conformal inclusions of Lie groups, the group of braided auto-equivalences has been computed in earlier works of the author [13,Theorem 1.1]. For completeness, we collect the results here.…”