Citation: Rowell, Eric, Richard Stong, and Zhenghan Wang. "On classification of modular tensor categories." Communications in Mathematical Physics 292.2 (2009): 343-389.
Web: https://arxiv.org/abs/0712.1377
Tags: Mathematical, Modular-tensor-categories, Property-F
This paper is one of the first to give a systemic list of all modular tensor categories of low rank, and is used as a very common reference for researchers in the field. Since this paper, there have been various other numerical searches which classify modular tensor categories of low rank:
> Bruillard, Paul, et al. "On classification of modular categories by rank." International Mathematics Research Notices 2016.24 (2016): 7546-7588.
> Ng, Siu-Hung, Eric C. Rowell, and Xiao-Gang Wen. "Classification of modular data up to rank 11." arXiv preprint arXiv:2308.09670 (2023).
> Alekseyev, Max A., et al. "Classification of modular data of integral modular fusion categories up to rank 11." arXiv preprint arXiv:2302.01613 (2023).
Similar numerical searches have been done in other settings, such as fusion categories. It is very useful to have a conception of what small rank examples exist, for a medley of reasons. One nice theme of these papers is that they also highlight some of the most important and non-trivial features of modular tensor categories. In particular, these papers often exploit the implicit number theory lying in the pentagon and hexagon equations.
The fact that a rank-by-rank classification of MTCs is possible follows from the rank finiteness theorem:
> Bruillard, Paul, et al. "Rank-finiteness for modular categories." Journal of the American Mathematical Society 29.3 (2016): 857-881.