Would exporting category lists be useful?

[Chenqitrg]

Hi developers,

It seems in artifact, there are functions list_fusionrings, list_fusioncategories, list_braidedcategories that can return lists of categories extracted from the file name.

So far I need to go though the data branch to see all possible solutions. Would exporting these informations to users be helpful? If so, would a documentation style of printing be helpful also? For example by providing more arguments in list_braidedcategories to allow print all possible categories with the given rank, or given central charge, or something else.

[borisdevos]

Hi Chenqi, for something like rank it should be easy, since that can be read off the types that define the categories. E.g. findall(x -> rank(x) == 2, CategoryData.list_fusioncategories()) can work. However, for something else like central charge, which is a property of the Object (read: sector), whatever function that will filter to this will have to calculate this value for every Object. Of course, we can store this somewhere, which should be easier now with the new TensorKitSectors release. Whether this gets stored in the list_ functions or not is a choice to be made, as we currently use these functions just for testing purposes.

[lkdvos]

I have no problem with exporting/marking these lists as public, and indeed it could be nice to try and generate some sort of documentation pages for this automatically.
For the filtering, I don't think it really makes sense to store this, as the number of combinations of things to query quickly explodes, and it is always possible to do something like:

mylist = [ring for ring in list_fusionrings() if predicate(ring)]

It could make sense though to try and add a large table to a docs page which just computes many of these?

[Chenqitrg]

I find it is helpful by running this code, which is still very dirty:

for C in CategoryData.list_braidedcategories()
       sec = Object{C}
       @show C
       @show ismodular(sec)
       if ismodular(sec)
       @show topological_central_charge(sec)
       end
       @show topological_spin.(values(sec))
end

Generating the table when using it is not slow.
Perhaps with following function?

list_braidedcategories(rank = R) # output all rank-R PMFCs
list_braidedcategories(rank = R, ismodular = M) # output all rank-R UMTCs
list_braidedcategories(rank = R, quantumdimension = M) # output all rank-R PMFCs with quantum dimension lists
list_braidedcategories(rank = R, Smatrix = true) # output all rank-R UMTCs with Smatrices

And adding the output to the documentation is also definitely helpful. For simplification, it is only possible to print some gauge-invariant data:

• For fusion ring, the data is only fusion rule and quantum dimensions. The quantum dimensions are secondary that diagonalize fusion matrices. A possible way is to give fusion rule aliases just like AnyonWiki, by using the most common fusion category of its categorification. Another approach is to output some "generators" of fusion rules, because most fusion rules are redundant and can be derived from several basic ones. We can also only output quantum dimensions for simplicity.
• For fusion category, besides quantum dimensions and fusion rule, we can export Frobenius-Schur indicators for self-dual objects. I do not know whether fusion category has some extra gauge invariant data. It seems to be a very deep mathematical field. I also have no idea how these UFC data was originally obtained, which is crazy! If it is possible, running the code again written by Jacob can fill the missing PMFCs. These quantum dimensions are algebraic numbers, do not know whether it is possible to directly use exact numbers in documentation.
• For braided category, it has many gauge invariant data and can almost uniquely determine a category when rank is small. They include: topological spins, topological central charge (these two are rational and can encode much information), quantum dimensions, S matrices, T matrices, modularity, Müger center (centralizier). Note that quantum dimension is not independent now because they are just first rows of S matrices. T matrices are the same as topological spins. For modular category, fusion rule can be determined by S matrix via Verlinde formula, thus is also not independent. Müger center is another gauge invariant data that determines how non-modular a braided category is. Most compactly, we can output quantum dimensions, topological spins, modularity and central charge. For more detailed output, we may replace quantum dimensions by S matrices, and also output Müger center.