List of Small Multiplicity-Free (Pivotal) (Braided) Fusion Categories

The list below contains all complex multiplicity-free (pivotal) (braided) fusion categories up to rank 7. The data on this page comes from the following thesis. All data on these categories can also be downloaded here, but note that the fusion categories link to fusion rings via their UUID, which can be obtained from the dataset on fusion rings.

The symbolic solutions are also available as part of the Anyonica and the TensorCategories.jl packages. Any category with formal name ${\text{FC}}_{d,e,f,g}^{a,b,c}$ can be obtained by evaluating the following code:

Anyonica:            FusionCategoryByCode[{a,b,c,d,e,f,g}]  (or FCBC[{a,b,c,d,e,f,g}])
TensorCategories.jl: anyonwiki(a,b,c,d,e,f,g)

Note that different packages might have different conventions and that, while Anyonica represents its data in a unitary gauge, not every package follows this convention.

Conventions

Every (pivotal) (braided) fusion category in the list comes with a specific pivotal structure (which exists for all our examples) and braiding (if it exists). Two (pivotal) (braided) fusion categories are regarded as equivalent only if they are equivalent as pivotal (braided) fusion categories. So even if two fusion categories have an equivalent associator (F-symbols), they will still be regarded as different if they are inequivalent as braided (if applicable) pivotal fusion categories.

Each category comes with a formal name ${\text{FC}}_{d,e,f,g}^{a,b,c}$ where $a,b,c,d$ are the labels coming from its fusion ring and $e,f,g$ label inequivalent associators, braidings, and pivotal structures respectively. If a fusion category admits no braiding then $f=0$.

The following symbols label the corresponding property of a fusion category. Right-click the table header to show or hide columns.

	$\text{Unitary}$	The category admits a unitary structure.
	$\text{Braided}$	The category admits a braiding.
	$\text{Spherical}$	The pivotal structure is spherical.
	$\text{Ribbon}$	The category is braided and spherical, hence ribbon.
	$\text{Modular}$	The category is ribbon with a non-degenerate S-matrix.
—	$\text{Pivotal}$	The category comes with a pivotal structure.