AnyonWiki and stored centers

AnyonWiki is a computational census of multiplicity-free pivotal fusion systems of rank at most seven. Its keys distinguish labeled solutions, including separate choices of associator, braiding, and pivotal structure; they do not merely name tensor-equivalence classes. The construction and organization of the census are described in Vercleyen (2024), §§4.4–4.5 and Appendix 9, and the archived data themselves are published as Vercleyen and Slingerland (Jun 2023).

anyonwiki(r,m,n,i,a,b,p) loads data from the package's pinned AnyonWiki artifact. The seven indices are:

The field-first form anyonwiki(K,r,m,n,i,a,b,p) is also available. Number-field, algebraic-closure, and finite-field inputs attempt scalar extension along a compatible field map. An AcbField input evaluates the stored exact category numerically, using precision(K) as the target approximation precision. The resulting category may use additional guard bits; its actual working precision is precision(base_ring(C)), as explained under Numerical fusion categories.

The available keys are those in the pinned dataset. The loader labels the simples $(\mathbb 1,X_2,\ldots,X_r)$ in dataset order; it does not retain more descriptive names from the source. A key identifies a stored solution with particular labels and bases, rather than a category up to tensor equivalence. The constructor reproduces the $F$- and pivotal coefficients from the record with this key in the pinned TensorCategoriesDatabase artifact, together with the $R$-coefficients when the braiding index is nonzero. Its stored dictionaries use :column_major_packing; the decoder transposes the packed arrays into the package's row-coordinate structural matrices. This reconstructs the pinned artifact record exactly after the stated coordinate conversion. The pinned artifact fixes the gauge of each entry; individual entries need not have separately named gauges in the literature.

anyonwiki_keys(n, attrs...) lists available keys of rank at most $n$; the default bound is $n=7$. The optional string filters are "spherical", "modular", and "unitary", and several filters may be supplied together:

anyonwiki_keys(5, "modular", "unitary")

using TensorCategories, Oscar
C = anyonwiki(2,1,0,1,1,1,1)
@assert length(simples(C)) == 2
@assert pentagon_axiom(C)
@assert hexagon_axiom(C)
(base_ring(C), dim.(simples(C)))

(Cyclotomic field of order 4, AbsSimpleNumFieldElem[1, 1])

The loader reads an exact coefficient field and a chosen complex embedding, together with $F$-symbols and pivotal data. When the braiding index is nonzero, it also reads $R$-symbols. It uses the dictionary decoder described in data exchange. Changing only the embedding changes numerical interpretation, whereas scalar extension requires an actual field map. An artifact's exact data need not coincide with a later revision of the website.

Precomputed centers

Drinfeld centers turn these input fusion categories into braided, and in the appropriate setting modular, categories. The stored presentations are the rank-at-most-five census computed by Mäurer and Thiel (2024), §7, rather than centers recomputed when the loader is called.

anyonwiki_center(r,m,n,i,a,b,p) loads a precomputed split skeletal braided fusion category. The stored centers are themselves fusion categories available directly from the package; calling the loader does not run the center algorithm. Each archive supplies its own exact coefficient number field, simple labels, and structural matrices, together with a complex embedding when one is stored; this field need not be the defining field of the corresponding AnyonWiki input. The current lookup covers input AnyonWiki categories with $r\leq5$; the rank of the resulting center can be larger. It does not search arbitrary input ranks or compute a missing entry automatically. The paper and Mäurer (2026), §5.1 describe the construction and census; the exact matrices for a particular key are fixed by the pinned artifact because the paper does not print every archived entry.

A saved split skeleton supplies fusion rules and structural matrices. By contrast, center(anyonwiki(...)) constructs the center over the same defining field as the loaded AnyonWiki category, and that center need not split over this field. Its objects explicitly carry half-braidings in the original category. Use this construction when those maps, rather than only $F$- and $R$-symbol data, are needed. Availability of one stored center does not imply that all AnyonWiki choices with the same fusion-ring index have a saved entry.