AnyonWiki

AnyonWiki Manual

Welcome to the comprehensive manual for AnyonWiki, your resource for fusion rings and fusion categories data.

What is AnyonWiki?

AnyonWiki is an open-source database containing comprehensive information about:

  • Multiplicity-free fusion rings up to rank 9
  • Fusion categories with complete mathematical data
  • Categorifiability criteria for various category types
  • Downloadable datasets including F-symbols, R-symbols, and more

The database serves researchers in topological quantum computation, category theory, and related fields.

Main Pages

Fusion Rings Database

Browse all multiplicity-free fusion rings organized by rank. Each entry includes:

  • Formal name and common name
  • Rank and Frobenius-Perron dimension
  • Multiplication table
  • Character table
  • Categorifiability properties

Explore Fusion Rings →

Fusion Categories Database

Explore fusion categories with detailed mathematical structures:

  • Pentagon and hexagon equation solutions
  • F-symbols and R-symbols
  • Pivotal and braiding structures
  • Associated fusion ring

Explore Fusion Categories →

Understanding the Data

Fusion Ring Naming

Fusion rings follow the formal naming convention:

FRnr,m\text{FR}^{r,m}_n

Where:

  • rr = rank (number of simple objects)
  • mm = multiplicity structure parameter
  • nn = serial number within the classification

Example: FR22,0\text{FR}^{2,0}_2 refers to the Fibonacci fusion ring.

Fusion Category Naming

Fusion categories use a seven-parameter code:

FCd,e,f,ga,b,c\text{FC}^{a,b,c}_{d,e,f,g}

These parameters encode the categorical structure and relate to the underlying fusion ring.

Formal Codes

Each fusion ring has a formal code array [rank, m1, m2, n] that uniquely identifies it:

  • rank: Number of simple objects
  • m1, m2: Multiplicity parameters
  • n: Serial number

Barcodes

A barcode is an alternative unique identifier for fusion rings, represented as an integer array.

Data Fields

For Fusion Rings

  • texnames: Array of alternative names in LaTeX notation
  • multTab: Multiplication table (fusion rules)
  • numericCharacters: Character table values
  • numericFrobeniusPerronDimensions: FP dimensions for each simple object
  • numericFrobeniusPerronDimension: Total DFP2\mathcal{D}_{\text{FP}}^2
  • tensorProductDecompositions: Tensor product data
  • subFusionRings: Sub-ring structures
  • references: Academic citations
  • software: Computational tools used

For Fusion Categories

  • fSymbols: F-symbols (6j coefficients)
  • rSymbols: R-symbols (braiding data)
  • pivotalStructure: Pivotal structure data
  • ribbonStructure: Ribbon category data (if applicable)
  • modularData: S and T matrices (for modular categories)

Categorifiability Properties

The database tracks whether fusion rings can be categorified to various types of categories:

Category Types

  • FC (Fusion Category): Basic fusion category structure
  • PFC (Pivotal Fusion Category): Admits pivotal structure with dual objects
  • UFC (Unitary Fusion Category): Admits unitary structure
  • BFC (Braided Fusion Category): Admits braiding (symmetric or not)
  • MFC (Modular Fusion Category): Non-degenerate braiding (modular tensor category)

A value of T (True) indicates at least one categorification exists with that structure.

Learn more about Categorifiability →

Searching and Filtering

Search Functionality

Use the search bar to find fusion rings or categories by:

  • Formal name (e.g., "FR22,0FR^{2,0}_2")
  • Common name (e.g., "Fibonacci", "Ising")
  • LaTeX notation

Filter Options

By Rank: Use the rank slider to filter fusion rings by number of simple objects.

By Categorifiability: Filter by specific category types (FC, PFC, UFC, BFC, MFC).

Sorting:

  • By rank (ascending/descending)
  • By DFP2\mathcal{D}_{\text{FP}}^2 value

Downloading Data

Individual Downloads

On each fusion ring or category detail page, click the download button to export:

  • JSON format with complete mathematical data
  • Includes all structure constants, characters, and metadata

Bulk Downloads

Download complete datasets:

Numeric Solutions:

  • All pentagon and hexagon equation solutions
  • F-symbols and R-symbols for all categories
  • Organized by fusion ring folders

Download Numeric Categories →

JSON Database:

  • Complete fusion rings database
  • All categorification data
  • Suitable for computational research

Using the Data

Software Integration

Anyonica (Mathematica)

FusionCategoryByCode[[a,b,c,d,e,f,g]]
(* or use shorthand *)
FCBC[[a,b,c,d,e,f,g]]

TensorCategories.jl (Julia)

anyonwiki(a,b,c,d,e,f,g)

Learn more about Software Tools →

File Format

Downloaded data files follow this structure:

Pentagon Equations (pentsol):

  • Tab-separated values
  • First 6 columns: object labels
  • Last 2 columns: Real and imaginary parts of F-symbol

Hexagon Equations (hexsol):

  • Similar format for R-symbols (braiding)

Pivotal Structures:

  • Pivotal element data for each object

Academic Use

Citing AnyonWiki

When using data from AnyonWiki in academic work, please cite:

  • The specific fusion ring or category by its formal name
  • The AnyonWiki database: https://anyonwiki.org
  • Original research papers listed in the references

References

Each entry includes:

  • Original research papers
  • DOI links where available
  • Software tools used in computation

Contributing

AnyonWiki is an open-source project. You can contribute by:

  • Submitting new fusion ring data
  • Adding categorification results
  • Improving documentation
  • Reporting errors or inconsistencies

Learn how to Contribute →

Support and Community

Getting Help

  • Documentation: Browse the full docs for detailed information
  • GitHub Issues: Report bugs or request features
  • Academic Contact: Reach out to the research team

Conventions Used

For detailed information about naming conventions, notation, and data structures:

View Conventions →