Cyclic extensions of fusion categories via the Brauer–Picard groupoid

Abstract

We construct a long exact sequence computing the obstruction space, \pi_1\mathrm {BrPic}(\mathcal C_0) , to G -graded extensions of a fusion category \mathcal C_0 . The other terms in the\linebreak sequence can be computed directly from the fusion ring of \mathcal C_0 . We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G -extensions. The most striking of these is a \mathbb{Z}/2\mathbb{Z} -extension of one of the Asaeda–Haagerup fusion categories, which is one of only two known 3 -supertransitive fusion categories outside the ADE series. In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology.