**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?*



Classification of unitary modular tensor categories (UMTCs)
1. Recently, Mignard and Schaunberg found a counter example to the conjecture (<https://arxiv.org/abs/1708.02796>). Counting $(S,T)$ pairs is still just a shorthand way of counting monoidal equivalence classes, which is not exactly the same as counting gauge classes (it's close though). Taking a note along that line though, it is known that for whatever fusion rules you want to study there exist polynomial invariants (<https://arxiv.org/abs/1509.03275>) which will get the job done at all levels from fusion to modular.
2. Eric Rowell showed (<https://arxiv.org/abs/math/0503226>) that the braided tensor categories coming from $\mathcal C(\mathfrak{so}\_{2r+1},l,q)$ for $l$, $2(2r+1)<l$, and $q^2$ a primitive $l$-th root of unity are not unitarizable (this implies no unitary solutions to pentagon and hexagon equations).
3. The categories above are modular if and only if $r$ is odd (see theorems 4.3 and 4.5 of Rowell).


This answer is related: [Is the representation category of quantum groups at root of unity visibly unitary?](https://mathoverflow.net/questions/156625/is-the-representation-category-of-quantum-groups-at-root-of-unity-visibly-unitar/156637#156637)


