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arXiv · 1411.2313

On the Classification of Weakly Integral Modular Categories

Paul Bruillard·César Galindo·Siu-Hung Ng·Julia Plavnik·Eric C. Rowell·Zhenghan Wang
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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 two actions on modular categories: 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.

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Paul Bruillard, César Galindo, Siu-Hung Ng, Julia Plavnik, Eric C. Rowell, Zhenghan Wang. 2014-11-10. On the Classification of Weakly Integral Modular Categories. https://arxiv.org/abs/1411.2313

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