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?


[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.


1 Answer 1


A detailed discussion of your question can be found in Davidovitch et. al's paper "On arithmetic modular tensor categories". They say that it is still an open problem whether there are non-cyclotomic modular tensor categories. However, they do conjecture formally (conjecture 4.5) that every MTC is cyclotomic. I am not aware of any significant recent progress on the conjecture.


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