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).


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?

  • $\begingroup$ I'm a bit confused about question 3. Do you still want modular fusion rules? Or are you relaxing the conditions on the fusion rules? $\endgroup$ Oct 29, 2017 at 2:55
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    $\begingroup$ My reading on this was that he's asking about fusion rules which lead, at most, to unitary premodular categories but no modular ones. $\endgroup$ Oct 29, 2017 at 17:06

1 Answer 1

  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?

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    $\begingroup$ A much better answer than I could have ever hoped for. Regarding (1), an interesting question then arises: Do we define anyonic theories as corresponding to (a) Gauge classes of UMTCs, (b) Monoidal classes of UMTCs, or (c) A single UMTC i.e. as distinguished by the described polynomial invariant? I suppose that practically/naively, (a) suffices, although Wang et al. use (b) (due to physical significance of the S-matrix) - but perhaps ultimately, it is defined as (c), and we only care to distinguish at the level of (a) or (b)? $\endgroup$ Oct 30, 2017 at 0:05
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    $\begingroup$ @S Valera I believe that the polynomials are supposed to be gauge invariant, so they don't distinguish more finely than (a). I would strongly suspect that (a) is the correct physical notion of equivalence for anyonic theories, though I can't prove it. $\endgroup$ Oct 31, 2017 at 1:44
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    $\begingroup$ A Grothendieck class refers to categories which share a set of fusion rules up to permutations of anyon types. A gauge class refers to distinct solutions to the pentagon/hexagons up to change of basis on the underlying hom spaces of the category (note: This is NOT the same thing as gauging in the physics literature). Monoidal classes can be seen as incorporating both of these things. $\endgroup$ Oct 31, 2017 at 1:46
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    $\begingroup$ As far as polynomial invariants for fusion categories go, there is a finite set of invariants which determine gauge classes and these can further be combined to obtain invariants which determine monoidal classes. For an example on how this is done see section 5.2 of 1509.03275. A more comprehensive example is sections 5 and 6 of 1608.03762. $\endgroup$ Oct 31, 2017 at 1:51
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    $\begingroup$ Anyonic theories correspond to monoidal classes of UMTCs. $\endgroup$ Oct 31, 2017 at 19:57

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