Citation: Mignard, Michaël, and Peter Schauenburg. "Modular categories are not determined by their modular data." Letters in Mathematical Physics 111.3 (2021): 60.
Web: https://arxiv.org/abs/1708.02796
Tags: Mathematical, Modular-tensor-categories, No-go
In this paper, the authors show that modular tensor categories are not determined by their modular data. In fact, there can be arbitrarily many modular tensor categories which all share the same modular data. The smallest example found is two non-equivalent categories of rank 49.
The examples constructed came from twisted group-theoretical modular tensor categories, the simplest kind. However, the categories which were shown to have the same modular data were still quite "similar" in a very real sense, and so one might expect that not much more data needs to be added to the modular data to get uniqueness. This effort of figuring out how to classify MTCs now that we know that modular data is insufficient has led to several follow-up works:
> Bonderson, Parsa, et al. "On invariants of modular categories beyond modular data." Journal of Pure and Applied Algebra 223.9 (2019): 4065-4088.
> Delaney, Colleen, and Alan Tran. "A systematic search of knot and link invariants beyond modular data." arXiv preprint arXiv:1806.02843 (2018).