Graded extensions of generalized Haagerup categories

Abstract

We classify certain ${\mathbb{Z}}_2$-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: ${\mathbb{Z}}_2$-graded extensions of ${\mathbb{Z}}_{2n}$ generalized Haagerup categories for all $n\le 5$; ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded extensions of the Asaeda-Haagerup categories; and extensions of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalized Haagerup category by its outer automorphism group $A_4$. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C${}^{\ast }$-algebras.