# Is every fusion category with unitary $S$ a UMTC?

A tensor category whose solutions to the pentagon/hexagon equations are unitary and whose braiding is nondegenerate is called a unitary modular tensor category (UMTC). When trying to find UMTCs, modularity demands that the relevant $S$-matrix is necessarily unitary (this guarantees nondegenerate braiding). Imposing this condition on $S$ helps to discard models which describe unphysical theories (a technique that is used e.g. by Wang et al. in https://arxiv.org/abs/0712.1377). From this, it is clear (and obvious through terminology) that:

$\{UMTCs\} \subseteq \{\text{fusion categories with unitary }$S$\}$

Note: From discussion in the comments below, it appears that fusion categories with unitary $S$ are called MTCs. I will word my question on this basis.

What I don't know is whether or not demanding $S$ unitary is necessary and sufficient for yielding a UMTC i.e. is it true that

$\{UMTCs\} = \{MTCs\}$ ?

Are there no counterexamples i.e. does demanding $S$ unitary always mean there exists a solution with $F$ and $R$ symbols unitary?

EDIT: Refer to https://mathoverflow.net/questions/284647/classification-of-unitary-modular-tensor-categories-umtcs for the refined version of this question.

• The wording of your question is a little bit confusing. By "anyonic tensor category", do you actually mean a UMTC which can be realized as the topological excitations of some Hamiltonian? The way your question is worded seems like it's asking a mathematical question about whether all unitary tensor categories are necessarily modular -- which is certainly not true. – Dominic Else Oct 27 '17 at 21:38
• Actually, I guess your question reads as asking whether every UMTC is a unitary tensor category -- which is vacuously true... – Dominic Else Oct 27 '17 at 21:58
• Essentially: does $S$ unitary always imply that its associated fusion rules have unitary solutions for $F$ and $R$ symbols? Or are there cases where this isn't true? – S Valera Oct 27 '17 at 22:59
• OK, so I guess you're asking whether every braided fusion category with unitary S matrix is unitary. I don't know the answer to that, but you had better edit your question, because the definition of UMTC by definition includes unitarity of F and R so the way you worded it is not correct. It's also really a math question, not a physics one, so might be worth asking on mathoverflow or math.stackexchange. – Dominic Else Oct 27 '17 at 23:12
• Yes, "unitary" means unitarity of F and R, "modular" means unitarity of S and T. – Dominic Else Oct 27 '17 at 23:50