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.