Classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories (with an appendix by Ryan Johnson, Siu-Hung Ng, David Penneys, Jolie Roat, Matthew Titsworth, and Henry Tucker)

Abstract

A unitary fusion category is called $\mathbb{Z}/2\mathbb{Z}$-quadratic if it has a $\mathbb{Z}/2\mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalisation of Ostrik’s results on formal codegrees to analyse the induction of the group elements to the centre, and a computation similar to Larson’s rank-finiteness bound for $\mathbb{Z}/3\mathbb{Z}$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.