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Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences.

(Spherical) fusion categories have well-known invariants like their global dimension, their rank, the categorical and Frobenius dimensions of their simple objects, or the Frobenius-Schur coefficients. Ribbon fusion categories have some more invariants like the $S$-matrix, the twist eigenvalues, and all the invariants of its symmetric centre. Graded fusion categories have e.g. the size of the group and the dimension as an invariant.

An example for something that is not an invariant is an $F$-matrix. It depends on the choice of basis vectors for trivalent morphism spaces, and that is not preserved by an equivalence. Summing over the appropriate elements of the $F$-matrices yields the Frobenius-Schur coefficients, though, and they are invariants.

A (spherical) $G$-crossed braided fusion category (short: $G\times$-BFC) is a $G$-graded fusion category with a compatible $G$-action and a crossed braiding. This implies e.g. that its trivial degree is a ribbon fusion category. All this gives us access to the invariants I've already mentioned, but I want to know whether a $G\times$-BFC has any new invariants.

Question. Are there any invariants of $G\times$-BFCs that are not invariants of the underlying $G$-graded fusion category, or of the trivial degree? For example, does the crossed braiding contain information beyond the trivial degree, and what information does the $G$-action possess?

Bonus question. Can those new invariants be expressed diagrammatically, i.e. in the graphical calculus of $G\times$-BFCs?

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Modular tensor categories give (via Reshetikhin-Turaev) a 321 oriented TFT. This gives a huge source of invariants, in particular any closed oriented 3-manifold gives a numerical invariant of MTCs. For example, the rank is the value of the 3-torus, while the global dimension is inverse of the square of the value of the 3-sphere. More elaborate invariants like the S-matrix and T-matrix also come up this way: the vector space assigned to the torus has a natural basis and the S-matrix and T-matrix are the value of the TFT on the mapping cylinders of the generators of the mapping class group of the torus.

To any spherical G$\times$-BFC you get a 321 G-HTFT, that is a TFT defined on oriented bordisms endowed with a map to BG. So completely analogously to the above you get an invariant of G$\times$-BFC assigned to any closed oriented 3-manifold endowed with a map to BG. Similarly you can get more elaborate invariants by looking at bordisms with boundary. I'm not sure which ones are the most useful, but that's a big source of invariants.

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  • $\begingroup$ Yes, this is sort of the direction I'm actually coming from. I'm starting with the TQFT (actually I'm looking at the 4d one) and I want to know what invariants I might get, so I have an idea on which manifolds I should calculate it. $\endgroup$ – Manuel Bärenz Aug 17 '18 at 20:41
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    $\begingroup$ There are some recent papers that use framed colored link invariants to distinguish some modular fusion categories with the same modular data: arxiv.org/abs/1806.03158 arxiv.org/abs/1805.05736 arxiv.org/abs/1806.02843 $\endgroup$ – zibadawa timmy Aug 18 '18 at 1:42
  • $\begingroup$ @zibadawatimmy, that's interesting, but that's only about modular categories, and not $G\times$-BFCs, right? $\endgroup$ – Manuel Bärenz Aug 21 '18 at 10:05
  • $\begingroup$ @ManuelBärenz Yes. I bring it up since knot complements are compact 3-manifolds, so since these are useful for distinguishing modular fusion categories, then ostensibly under Noah's analogous construction for the G-crossed situation they should be useful there as well. Though I'm not exactly the right person to ask how/if that works out; the papers I linked are already pushing the limits of my understanding on these topics. I think they should at least suffice to justify an expectation that these invariants hold new/interesting information, and point out specific ones with known usefulness. $\endgroup$ – zibadawa timmy Aug 21 '18 at 10:22
  • $\begingroup$ Let me rephrase. What 3-manifolds will give a new invariant of $G\times$-BFCs that is not just an invariant of the premodular category in its trivial degree, and a $G$-homotopy invariant (like Dijkgraaf-Witten). $\endgroup$ – Manuel Bärenz Aug 21 '18 at 10:43
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This is not a direct answer to the question, but rather a longer comment.

My claim is most of the information is contained in the $G$-extension of the trivial degree BFC.

Let $\mathcal F=\bigoplus_{g\in G} \mathcal F_g$ be a $G\times$-BFC and assume that the braiding on $\mathcal F_e$ is non-degenerate and that the $G$-grading is faithful. Then the Drinfel'd center of $\mathcal F$ is braided equivalent to $\mathcal F^G\boxtimes \mathcal F_e^{\mathrm{rev}}$ by DMNO and the fact that $(\mathcal F^G)_G\cong \mathcal F$. Additionally, there are central functors $\tilde\iota\colon\mathcal F_e^\mathrm{rev}\to \mathcal F$ and $\mathcal F^G\to \mathcal F$. The $G\times$-BFC can be recovered from the central functor $\tilde\iota$. As a tensor functor $\tilde\iota$ is just the embedding $\iota\colon\mathcal F_e\hookrightarrow \mathcal F$, which is determined by the $G$-graded structure of $\mathcal F$ as a fusion category.

So the only new additional information which $\mathcal F$ as a $G\times$-BFC has, is the braiding on $\mathcal F_e$ and the above central structure on the embedding $\mathcal F_e\hookrightarrow \mathcal F$. In many cases, this structure is unique.

But, I am not sure which new invariants the central structure gives. The only thing which comes to my mind is the "crossed" S-matrix. And I am not even certain if this is a true invariant of the $G\times$-BFC, considering my above discussion.

BTW, I showed that such a central structure always yields a $G$-crossed braided extension in the unitary setting [Prop. 2.4, https://arxiv.org/abs/1803.04949], namely:

Let $\mathcal D$ be a unitary modular tensor category and $\mathcal F=\bigoplus_g \mathcal F_g$ a (faithfully) $G$-graded unitary extension of $\mathcal F_e=\mathcal D$ together with a central structure $\iota\colon \mathcal D^\mathrm{rev}\to \mathcal F$ on the canonical inclusion functor. Then $\mathcal F$ is naturally a $G$-crossed braided extension of $\mathcal D$.

In the degenerate case, information seems to be lost. For example $\mathrm{Vec}$ is a $G\times$-BFC with $\mathrm{Vec}^G=\mathrm{Rep}(G)$, but $Z(\mathrm {Vec})\cong \mathrm {Vec}$ so the information about the equivariantization is not contained in the fusion category anymore, if you want.

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  • $\begingroup$ What's the "crossed" $S$-matrix? The double braiding isn't an endomorphism in a $G\times$-BFC. $\endgroup$ – Manuel Bärenz Aug 17 '18 at 20:39
  • $\begingroup$ There is something in the literature, IIRC. But I have not studied it in detail. Anyways, if you are interested in 4D TQFT the interesting examples are apparently (from what I understood from talks from Zhenghang Wang) when the grading is not faithful. Thus in this case my comments are not very useful, they consider only the case arising in 3D TFT and rational CFT. $\endgroup$ – Marcel Bischoff Aug 17 '18 at 20:44
  • $\begingroup$ Can you point me to the literature on a "crossed" $S$-matrix? I asked about something related here: mathoverflow.net/questions/290991/… Yes, I'm interested in the general case. Is there no generalisation of the central functor situation? $\endgroup$ – Manuel Bärenz Aug 17 '18 at 20:47
  • $\begingroup$ I have to check the literature myself. Maybe I remember something wrong. There is still a central functor. But I have never thought about it in detail. All the techniques use that the braiding is on-degenerate. $\endgroup$ – Marcel Bischoff Aug 17 '18 at 20:51
  • $\begingroup$ You say that the interesting examples are when the grading is not faithful. Is there any video of a talk where Wang claims this? $\endgroup$ – Manuel Bärenz Aug 21 '18 at 11:06

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