In a recent talk (see [1]) Richard Ng asks whether the invariants of 3-manifolds derived from any modular fusion category are cyclotomic integers. In tqft, such an invariant is computed from the F-symbols of a modular fusion category. The F-symbols depend on the choice of bases for the hom spaces, but the invariant is independent of the choice (see the book [2] by Vladimir Turaev and Alexis Virelizier).
A fusion category is called cyclotomic if there exist bases of the hom spaces for which the F-symbols are all cyclotomic numbers. In [3] Scott Morrison and Noah Snyder provided non-cyclotomic fusion categories, namely the principal even part of the Haagerup and extended Haagerup subfactors; they also proved that the Drinfeld center of the first one is cyclotomic (and they expect so for the second, but it is still unknown).
Question 1: Is there a non-cyclotomic modular fusion categories?
It is a weaker version of Richard's question mentioned above, because a negative answer to Question 1 answers Richard's question positively, whereas a positive answer to Question 1 let Richard's question open (because a sum of non-cyclotomic numbers can be cyclotomic).
The Drinfeld center of the Extended Haagerup category (mentioned above) is a candidate, but it is expected to be cyclotomic. This leads to the following question:
Question 2: Is the Drinfeld center of a fusion category always cyclotomic?
References
[1] R. Ng, Arithmetic of Modular Tensor Categories, Dec. 2021, https://www.math.ucsc.edu/seminars/colloquium.html#dec2
[2] V. Turaev, A. Virelizier, Monoidal categories and topological field theory. Progress in Mathematics, 322 (2017) xii+523 pp.
[3] S. Morrison, N. Snyder, Non-cyclotomic fusion categories. Trans. Amer. Math. Soc. 364 (2012), no. 9, 4713--4733.
