Classification of Integral Modular Categories of Frobenius–Perron Dimension pq 4 and p 2 q 2

Abstract

We classify integral modular categories of dimension pq 4 and p 2 q 2 , where p and q are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension 4q 2 . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension 4q 2 is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.