Possible modular categorification of FR^{7,2}_11

[aatmunbaxi]

The zesting construction appears to give a categorification of the fusion rules $\text{FR}^{7,2}_{11}$. Specifically there is a zesting of $\text{SU}(2)_6$ that gives a new braided (in fact, ribbon and modular) category with 2 non self-dual simple objects. See Example 5.7 in Braided Zestings of Verlinde Modular Categories and Section 6.3 of Braided Zesting and its Applications (arxiv version).

[gert-vercleyen]

Thanks for letting me know! I'm going to look into it.

[gert-vercleyen]

I looked at the log files and there is a stage where solutions were not accepted without throwing a warning or giving a reason why not. I will investigate this further.

[gert-vercleyen]

I finally found the error in the solver! 🥳

At some point I use tetrahedral symmetries to find zero values. I assumed that these symmetries map non-zero F-symbols to non-zero F-symbols and likewise zero F-symbols to zero F-symbols.
I can prove this is the case for 2 generators of the symmetries (corresponding to rotations) and I mistakenly assumed this was the case for the 3rd one (corresponding to a reflection) as well.

This is not the case. For FR^{7,1,2}_{11} there are only 2 possible sets of 0 values for the F-symbols (checked independently using PicoSAT): all symbols are non-zero (doesn't lead to a solution) or a set of 16 symbols are 0. These 16 are invariant under the rotational tetrahedral symmetries but not under the reflection.

The set of 16 F-symbols that are 0 was incorrectly rejected by the previous versions of the solver and so no solution was found.
I reduced the system of equations under the assumption that those 16 symbols are 0 and found multiple solutions.

Sadly (for me), this means that I will have to check the 0-values found for every ring and see whether there might be more categories. 😢

@aatmunbaxi Thank you so much for checking the data! It took me quite some time to figure out what went wrong and I would not have found out if you didn't question the results :)

It might take a week or more before the new cats are uploaded, though. It would be much more efficient to do an investigation in all the other ones first and upload all the new ones (hopefully not too many) in one go

[gert-vercleyen]

I found

• 4 non-equivalent solutions to the pentagon equations,
• 4 non-equivalent solutions to the hexagon equations per solution to the pentagon equations,
• 1 non-equivalent solution to the pivotal equations per solution to the pentagon equations

All categories are modular and 2 solutions to the pentagon equations lead to 8 unitary categories.

so in total there are 16 braided modular fusion categories with FR^{7,1,2}_{11} fusion rules of which 8 are unitary

There is 1 funny thing though: the SU(2)_6 fusion ring has

• 2 non-equivalent solutions to the pentagon equations
• 4 non-equivalent solutions to the hexagon equations per solution to the pentagon equations,
• 2 non-equivalent solutions to the pivotal equations per solution to the pentagon equations

this pattern is different from the solutions for the zested case. Moreover, the non-equivalent pivotal structures mess up unitarity for 4 of the fusion categories so SU(2)_6 only has 4 unitary fusion categories.

@aatmunbaxi Is this in line with the predictions from the zesting procedure?

[aatmunbaxi]

For SU(2)_6, zesting turns out to be insufficient to predict all braidings on the fusion ring. According to the parameterization of braided zests, we would see at most 2 pentagon solutions, only 2 braidings per pentagon, and 2*2 ribbon zestings producing modular categories. However, it is known by Kazhdan, Wenzl Reconstructing Monoidal Categories and Feng, Ming, Rowell Reconstructing braided subcategories of SU(N)_k that there are 2N braidings on the fusion ring SU(N)_k. So this seems to agree.

Regarding FR^7,2_11, zesting would similarly predict at most 2 pentagon solutions given fixed fusion rules, with 2 braidings per solution to the pentagon. It's likely that zesting underparameterizes braidings in this case too. Possibly techniques used in the second reference can be used to get these. The number of pentagon solutions is puzzling; usually one does this by choosing a cohomology class in H^3(Z_N,C^x), which would only give 2 solutions up to monoidal equivalence. Though this is for the actual SU(2)_6 ring and not the zested one. Perhaps up to an appropriate notion of equivalence it's okay.