On the classification of weakly integral modular categories

Abstract

We classify all modular categories of dimension 4m, where m is an odd square-free integer, and all ranks 6 and 7 weakly integral modular categories. This completes the classification of weakly integral modular categories through rank 7. Our results imply that all integral modular categories of rank at most 7 are pointed (that is, every simple object has dimension 1). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks 6 and 7 have dimension 4m, with m an odd square free integer, so their classification is an application of our main result. The classification of rank 7 integral modular categories is facilitated by an analysis of actions on modular categories by two groups: the Galois group of the field generated by the entries of the S-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.

On the Classification of Almost Square-Free Modular Categories

“…C is of type (1, 4; 2, 3) and hence it is a modular category of rank 7. By [2,Theorem 5.8], it should be pointed, a contradiction.…”

Dimension as a quantum statistic and the classification of metaplectic categories

“…We will disregard this case, pausing only to ask: We can consider C prime (i.e. not of the form C 1 ⊠ C 2 ) since otherwise it reduces to known cases, see [6,9].…”

“…As in the proof of Theorem 3.1 in [6], using the de-equivariantization process, we can assume that |U (C)| = 2 k , for k an integer number.…”

Classification of rank 5 premodular categories

“…Applying the Orbit-Stabilizer Theorem, we can conclude that |Z 3 .y| = |Z 3 .z| = 3 and C Z 3 is a rank 7 modular category of global dimension 1 + 3a 2 + 3b 2 where a = d 3 /3 and b = d 4 /3. So either a = b = 1 by [6] or we may apply Lemma 3.8 to conclude that a = 1. Applying the equidimensionality of the universal grading of C Z 3 , [24,8] we can deduce that b = 1 or 2.…”

“…Applying the equidimensionality of the universal grading of C Z 3 , [24,8] we can deduce that b = 1 or 2. The later case cannot occur as every integral modular category of rank 7 is pointed [6,Theorem 5.8]. Thus the global dimension of C is 21.…”