What is the corresponding UMTC of the double braided reversed Semion?

[Chenqitrg]

Hi,

I am trying to locate all low-rank UMTCs by rank and chiral central charge. However, at the rank 4, I encounter some problem. To my understanding of the PMFC convention, the category index labels all solutions of the pentagon equation and the braid index labels all solution of the hexagon equation when the solution of the pentagon equation is fixed. Therefore, in rank 4, for UniqueFusion category, if the self-duality number is zero, then the fusion ring solution is uniquely fixed to be $\mathbb{Z}_2 \times \mathbb{Z}_2$. Therefore we need to consider categories of the form PMFC{4(rank 4), 1(no multiplicity), 0(all objects are self-dual), 1(group-like fusion ring), x(some solutions of the pentagon equation), y(some solutions of the hexagon equation)}.

The trivial solution of the pentagon equation should at least include the following ones, when x = 0:

1. $\mathrm{Rep}(\mathbb{Z}_2 \times \mathbb{Z}_2)$
2. $\mathrm{Rep}(\mathbb{Z}_2) \boxtimes \mathrm{sVec}$
3. $Z(\mathrm{Rep}(\mathbb{Z}_2))$

Indeed, I find there are three categories PMFC{4, 1, 0, 1, 0, 0}, PMFC{4, 1, 0, 1, 0, 1} and PMFC{4, 1, 0, 1, 0, 2} in CategoryData. These three exactly matches the above three categories.

When we choose the second solution to the pentagon equation, we should at least have the following ones, when x = 1:

1. $Z(\mathrm{Semion})$
2. $\mathrm{Semion} \boxtimes \mathrm{Semion}$
3. $\mathrm{Semion} \boxtimes \mathrm{sVec}$
4. $\overline{\mathrm{Semion}} \boxtimes \overline{\mathrm{Semion}}$

However, there are only three solutions PMFC{4, 1, 0, 1, 1, 0}, PMFC{4, 1, 0, 1, 1, 1} and PMFC{4, 1, 0, 1, 1, 2}, corresponding to the above first three categories. I cannot find the PMFC{4, 1, 0, 1, 1, 3}, which may correspond to the 4-th one.

Edit: For trivial associator with $\mathbb{Z}_2 \times \mathbb{Z}_2$ fusion rule, there exists another three-fermion UMTC, whose $c = 4 \mod 8$. Would it be PMFC{4, 1, 0, 1, 0, 3}? But there is no such label.

Edit: Now I don't think $\mathrm{Semion} \boxtimes \mathrm{sVec}$ shares the same F-symbol with $\mathrm{Semion} \boxtimes \mathrm{Semion}$ and $\overline{\mathrm{Semion}} \boxtimes \overline{\mathrm{Semion}}$. They are definitely different as $H^3(\mathbb{Z}_2 \times \mathbb{Z}_2, U(1))\simeq \mathbb{Z}_2^3$. But still very confused. The twist should be a gauge-independent data.

[Chenqitrg]

Or is there an another possibility, that the third choice is actually the fourth one, but with some minor typo in R symbol? Because the 1,2,4 cases are the same UFC, and the only difference between them is the braiding data.

When I calculate topological spins of PMFC{4, 1, 0, 1, 1, 2}, I find they are $0, 1/4, 1/2, -1/4$. The correct one should be $0, -1/4, 1/2, -1/4$.

The reference I am using is the paper 1506.05768. It gives tables of low-rank UMTCs. These UMTCs can also be found in Kitaev's 16-fold way in his classic work.

[lkdvos]

I'm not sure if I can help you here, but just to let you know, our data is stored here: https://github.com/QuantumKitHub/CategoryData.jl/tree/data/data
This might be an easier way of verifying the results.
I don't know enough about this to tell if we have a mistake, this is a gauge choice, or the labels just don't match, but I can confirm that this is indeed the meaning of the labels, where the x and y are really just integer labels that came from "the order in which they were computed" which is completely random.
The approach of finding the right category then indeed should be to first identify the correct solutions of the pentagon, and then the solutions of the hexagon.
I'm not claiming that I am 100% sure there are no mistakes in the data, but on the other hand we are testing pentagon and hexagon equations for all of them, so this would be somewhat surprising (although I don't think we really are exhaustively checking everything, so don't take my word for that...).
I do apologize for not being able to better help you with this, but this is mostly the limit of my understanding...
Again, if you can identify some of these categories and provide aliases, that would be very appreciated.

[Chenqitrg]

Hi Lukas, thanks for your answer.
I will temporarily skip these parts and finish other parts first. I will try to finish below than rank 5, and submit a PR first and we can check further. There are too many UMTCs in rank 6 and rank 7.
A good information of rank 6 UMTCs is: most of them are non-primitive. Therefore they are not that complicated.

[Chenqitrg]

I indeed tested the hexagon equation for PMFC{4, 1, 0, 1, 1, 2} and there is indeed no error.

[Chenqitrg]

It seems the pentagon equation of $\mathbb{Z}_2 \times \mathbb{Z}_2$ contains fewer than it should be. I looked up the file Fsymbols, and find there are only four solutions from FR_4_1_0_1_0 to FR_4_1_0_1_3. In theory there should be 8 of them.

[lkdvos]

Are these all unitary? I think we only have the unitary ones

[Chenqitrg]

Yes, they are all unitary. Generator can be find in Eq.(73) in /1706.09769

[JCBridgeman]

Hi @Chenqitrg. It's been a very long time since I collected these solutions, so it's very possible that I've missed some. I also don't remember too well all of the details.

Could you let me know one of the categorifications we've missed (and how to understand it).
Note that we only provide solutions up to the following equivalence: If the 2 solutions are related via a permutation of simple objects + gauge transformation, you'll only find 1 of those.

For example, in the Z2xZ2 case, there are apparently 8 solutions, corresponding to choosing a FS-indicator for the 3 non-trivial objects, however these can be pair-wise exchanged .

[Chenqitrg]

Thank you for the clarification! I apologize for missing such a possibility. I still need to further check the case PMFC{4, 1, 0, 1, 1, 2}. It seems to me the satisfaction of the hexagon equation and twist 0, 1/4, 1/2, -1/4 cannot be simultaneously satisfied.

There may be some missing braided solution for {4,1,0,1,0}, {4,1,0,1,1}, {5,1,0,3,0}, {5,1,0,3,1}. I will try to test them further. Some already located UMTCs are listed in a draft PR #41 .