**Context/background:**

I'm approaching this topic from the perspective of anyonic systems. In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that

i) The category is *unitary* (all $F$ and $R$ symbols are unitary) i.e. a *UTC*.

ii) The category is *modular* ($S$ matrix is unitary - this ensures nondegenerate braiding) i.e. an *MTC*.

Therefore, we are interested in classifying all *UMTCs* (which may play host to a physical theory of anyons in Nature). The advantage of working with MTCs is that demanding $S$ unitary provides a nice mechanism for discarding fusion rules that are unphysical (a technique that is used by *Wang et al.* in https://arxiv.org/abs/0712.1377).

**Questions:**

*1) Given modular fusion rules $\mathcal{N}$, how do we count the number of corresponding UMTCs?*

The obvious way seems to be to count the no. of distinct unitary solutions of $F$ and $R$ symbols (upto gauge freedom). However, it is *conjectured* that the $S$ and $T$ matrices uniquely define a UMTC, and so we might count the number of 'modular symbols' $(\mathcal{N}; S, T)$ corresponding to $\mathcal{N}$ - only this increases the count due to symmetries $(\mathcal{N}; -S, T)$, $(\mathcal{N}; S^{\dagger}, T^{\dagger})$ of the symbol. Which way is the more appropriate method for counting UMTCs w.r.t $\mathcal{N}$? The latter method seems stronger yet based on conjecture (though I suppose this is harmless in practice).

*2) Are there any known examples of modular $\mathcal{N}$ with no unitary solutions for $F$ and $R$ symbols?*

*3) Are there examples of fusion rules $\mathcal{N}$ that yield UTCs which aren't modular?*

modularfusion rules? Or are you relaxing the conditions on the fusion rules? $\endgroup$