List of Small Multiplicity-Free Fusion Categories

This page is still in construction.

All numeric solutions of the pentagon and hexagon equations for all multiplicity free fusion categories up to and including rank 7 can be downloaded from the following link

• Numeric solutions

The numeric solutions are organized as follows. For each fusion ring there is a folder that contains several files named pentsol_i.txt and hexsol_i_j.txt. The index i labels a solution to the pentagon equations and the index j labels a solution to the hexagon equations with $F$-symbols coming from pentsol_i.txt. The pentsol files contain rows of tab-separated numbers of which the first 6 correspond to the anyon labels and the last two to the real and imaginary parts of the F-symbol. The same format is used for the hexsol files. Every folder contains a file with structure constants that define the labeling of the particles so one can find out how to permute the labels of the solutions, if necessary. Normally all rings and categories in these files should be labeled the same way as on this web site.

The symbolic solutions are available as part of the Anyonica package

• Anyonica

but do take a look at the Software page that contains other sources and amazing software as well!

In the future we plan to:

• create some pages that give a clearer overview of the solutions
• add the solutions to the pages of the fusion rings
• Keep adding categories, of course :)

List of multiplicity-free fusion categories up to rank 7

Below are the multiplicity-free fusion categories up to rank 7. These should be all the solutions, both unitary and non-unitary, braided and non-braided, and for all admissible sets of possible $0$-values for the $F$-symbols. If any results seem odd, please raise an issue on the github page.