@CharlesRezk Good question. It is certainly known that you get a spectrum with homotopy groups: ?? Z/2 Z/2 C^times and that the top part is essentially a chunk of the Brown-Comenetz dual of the sphere (using C^\times in place of Q/Z). I am fairly certain that the bottom homotopy group (??) is trivial, just as in the non-super case. In fact I think the same argument works using a super version of global dimension, but I am not 100% sure. The literature on super tensor categories is less extensive.

It would be good to get that sorted out.

@ChrisSchommer-Pries is a rigid super tensor category a rigid monoidal category enriched over sVect, or is there some kind of Z/2-grading on objects (or is it something else entirely)?

@ChrisSchommer-Pries Ok. It seems disappointing if (??) is really trivial.

@ArunDebray I would define it as a monoidal object in sVect-module categories, which is also rigid. So it is a rigid monoidal category which is also tensored over sVect and where the monoidal structure is compatible with the sVect tensoring. With the usual finiteness hypotheses it will automatically be enriched over sVect as well. There is no Z/2 grading on objects in general, but tensoring with an odd line gives an interesting operation.

@CharlesRezk Yes, I agree it is disappointing. I take it as one more piece of evidence that this sort of "2-Vect" is too rigid. There are other problems too. For example the theory of 2-representations in this target works okay for finite groups, but fails fantastically for Lie groups (or things like the String group). Also we know that (most versions of) quantum Chern-Simons theory cannot be realized as extended TQFTs with this target.

On the other hand, there should be another 3-category which is very similar but looks quite a bit more promising using the "Bicommutant Categories" of Henriques-Penneys arxiv:1511.05226. I don't know if the whole Morita 3-category has been fully worked out, but it looks like when it is worked out it will receive a map from the 3-cat confromal nets. This means that pi_0 of the Picard 3-category has to be much more interesting!

From what we know about conformal nets, it forces pi_0 of this Picard 3-category to have a copy of Z/24nZ where n is an integer (possibly zero). If we knew exactly what pi_0 of the Picard category of conformal nets was, we could possibly constrain this further.