Fusion Categories

A fusion category is a mathematical structure that provides a categorical framework for studying topological phases of matter, quantum field theories, and anyon models. It categorifies the notion of a fusion ring.

Definition: Fusion Category

A fusion category $\mathcal{C}$ is a $\mathbb{C}$ -linear, semisimple, rigid monoidal category with:

Finitely many simple objects up to isomorphism
Simple tensor unit: $\mathbf{1}$ is simple
Finite-dimensional Hom spaces: $\dim \text{Hom}(X,Y) < \infty$
Every object has finite length

Key Components:

Objects: $\text{Obj}(\mathcal{C})$ with simple objects $\{X_0 = \mathbf{1}, X_1, \ldots, X_{r-1}\}$
Morphisms: $\text{Hom}_{\mathcal{C}}(X,Y)$ are $\mathbb{C}$ -vector spaces
Tensor product: $\otimes: \mathcal{C} \times \mathcal{C} \to \mathcal{C}$
Duals: Each object $X$ has a dual $X^*$

Structure Data

F-Symbols (6j Coefficients)

The F-symbols encode the associativity of the tensor product. They satisfy the pentagon equation:

\sum_{\mu} F^{ijk}_{\ell}{}^{m\mu} F^{m \ell n}_{p}{}^{ij\nu} = \sum_{\rho} F^{k \ell n}_{p}{}^{i\rho} F^{ijn}_{p}{}^{k\nu\rho}

These are complex numbers $F^{abc}_d \in \mathbb{C}$ that define natural isomorphisms:

(X_a \otimes X_b) \otimes X_c \xrightarrow{F} X_a \otimes (X_b \otimes X_c)

R-Symbols (Braiding)

For braided fusion categories, the R-symbols encode the braiding. They satisfy the hexagon equations:

R^{ab}_c \in \mathbb{C}

These define natural isomorphisms:

X_a \otimes X_b \xrightarrow{R} X_b \otimes X_a

Grothendieck Ring

The Grothendieck ring $K_0(\mathcal{C})$ of a fusion category is a fusion ring with:

Basis: $[X_0], [X_1], \ldots, [X_{r-1}]$ (isomorphism classes)
Multiplication: $[X_i] \cdot [X_j] = \sum_k N_{ij}^k [X_k]$

where $N_{ij}^k$ are the fusion coefficients from:

X_i \otimes X_j \cong \bigoplus_k (X_k)^{\oplus N_{ij}^k}

Types of Fusion Categories

Pivotal Fusion Category (PFC)

A fusion category with a pivotal structure - a monoidal natural isomorphism:

j: \text{Id}_{\mathcal{C}} \to (\ )^{**}

This gives:

Left and right duals coincide
Categorical dimensions: $\dim(X) \in \mathbb{C}$
Quantum dimensions: $d_i = \dim(X_i)$

Spherical Fusion Category

A pivotal category where left and right traces coincide. The pivotal structure is spherical if:

\text{tr}_L(f) = \text{tr}_R(f)

for all morphisms $f: X \to X$ .

Unitary Fusion Category (UFC)

A fusion category with a unitary structure:

$\mathbb{C}$ -linear structure extended to $*$ -structure
$\text{Hom}(X,Y)$ are Hilbert spaces
Tensor product is unitary
Quantum dimensions $d_i > 0$ are real

Physical significance: Models physical anyon systems

Braided Fusion Category (BFC)

A fusion category equipped with a braiding:

c_{X,Y}: X \otimes Y \xrightarrow{\sim} Y \otimes X

satisfying naturality and hexagon axioms.

Types of braiding:

Symmetric: $c_{Y,X} \circ c_{X,Y} = \text{id}_{X \otimes Y}$
Non-degenerate: Related to modular categories

Ribbon Fusion Category

A braided pivotal category with a twist (ribbon structure):

\theta_X: X \to X

satisfying:

$\theta_{\mathbf{1}} = \text{id}_{\mathbf{1}}$
$\theta_{X \otimes Y} = c_{Y,X} \circ c_{X,Y} \circ (\theta_X \otimes \theta_Y)$

Modular Fusion Category (MFC)

A braided fusion category where the S-matrix is non-degenerate:

S_{ij} = \text{tr}(c_{X_j, X_i} \circ c_{X_i, X_j})

Properties:

S-matrix: $(r \times r)$ symmetric matrix
T-matrix: $T_{ij} = \delta_{ij} \theta_i$ (diagonal)
$S^2 = C$ (charge conjugation)
$(ST)^3 = S^2$

Physical significance: Describes (2+1)D topological phases

Categorical Dimensions

Quantum Dimensions

For a pivotal category, the quantum dimension of $X_i$ is:

d_i = \dim(X_i) \in \mathbb{C}

In unitary categories: $d_i \in \mathbb{R}_{>0}$

Global Dimension

The global dimension (categorical dimension) is:

\mathcal{D}^2 = \sum_{i=0}^{r-1} d_i^2

For spherical categories, this equals $\mathcal{D}_{\text{FP}}^2$ (Frobenius-Perron dimension).

Formal Naming Convention

Fusion categories in AnyonWiki use the convention:

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

This seven-parameter code uniquely identifies the category and encodes:

Associated fusion ring information
Categorical structure type
Serial number in classification

Pentagon and Hexagon Equations

Pentagon Equation

The F-symbols must satisfy coherence (pentagon equation) for all objects:

F^{jkl}_n F^{inl}_m = \sum_p F^{ikl}_p F^{ijp}_m F^{jkl}_p

Computational challenge: System of polynomial equations in F-symbols

Hexagon Equations

For braided categories, R-symbols satisfy (with F-symbols):

\text{(Hexagon I and II equations)}

These ensure braiding is compatible with associativity.

Categorification

Question: Does a given fusion ring arise from a fusion category?

Answer: Not always! Some fusion rings cannot be categorified.

Obstruction Theory

Obstacles to categorification include:

Pentagon equation has no solutions
Solutions exist but not unitary
Gauge freedom complications

Learn more →

Data in AnyonWiki

For each fusion category, the database may contain:

F-symbols: Pentagon equation solutions
R-symbols: Hexagon equation solutions
Pivotal structure: Pivotal element data
Quantum dimensions: Numerical values
S-matrix: For modular categories
T-matrix: For modular categories

File Formats

Pentagon solutions (pentsol):

Tab-separated: i j k l m n Re(F) Im(F)
First 6 columns: object labels
Last 2 columns: Real and imaginary parts

Hexagon solutions (hexsol):

Similar format for R-symbols

Important Examples

Pointed Categories

Categories where all simple objects are invertible:

$\mathcal{C}(\mathbb{Z}_n)$ - Cyclic group
$\mathcal{C}(G)$ - Finite group $G$

$\text{Rep}(G)$

Category of finite-dimensional representations of finite group $G$ :

Rank = number of conjugacy classes
Braided if $G$ is abelian
Modular only for $G = \{e\}$

Quantum Group Categories

From quantum groups $U_q(\mathfrak{g})$ :

$\mathcal{C}(su(2)_k)$ - $SU(2)$ level $k$
Contains $k+1$ simple objects
Modular for all $k \geq 1$

Fibonacci Category

Rank 2, non-pointed
Universal for topological quantum computation
$\mathcal{D}^2 = \phi^2 + 1$ (golden ratio squared)

Ising Category

Rank 3
Describes Majorana fermions
Basis: $\{\mathbf{1}, \sigma, \psi\}$

Software Tools

Computing Category Data

Anyonica (Mathematica):

FusionCategoryByCode[[a,b,c,d,e,f,g]]

TensorCategories.jl (Julia):

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