Cyclic extensions of fusion categories via the Brauer-Picard groupoid

TL;DR

This paper develops a method to compute obstructions for G-graded extensions of fusion categories using the Brauer-Picard groupoid, leading to new examples and revealing deep connections with algebraic topology.

Contribution

It introduces a long exact sequence for obstruction spaces in fusion category extensions and applies it to construct new fusion categories from subfactors.

Findings

Constructed a long exact sequence for obstruction spaces in G-extensions.

Produced new fusion categories, including a Z/2Z-extension of an Asaeda-Haagerup category.

Linked the sequence to homotopy groups of a fibration, highlighting topology-fusion interplay.

Abstract

We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of 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 Z/2Z-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.