Anyons, CFT, and tensor-category language

The categorical language follows Etingof et al. (2015). The anyon conventions and fusion-tree language follow Bonderson (2007), Chapter 2.

Translation of terminology

In a unitary anyon model one usually works with a unitary braided fusion category equipped with compatible duality and spherical or ribbon data. The general package interface does not assume that every category has all these structures.

For a split fusion category, the integers $N_{ab}^{c}$ are the structure constants of its Grothendieck ring. Over a non-splitting field, simple endomorphism division algebras enter the multiplicity formula. The distinction between categorical and Frobenius–Perron dimensions also matters when the chosen spherical dimensions are not the positive Frobenius–Perron dimensions.

First anyon computation: Ising

The default constructor uses $\mathbb Q(\sqrt2)$, which contains the associator and pivotal coefficients needed below but not the phases of a braided realization. The Ising catalogue entry describes the coefficient-field choices. The package labels its three simple objects $(\mathbb 1,\chi,X)$; in the example we rename these as the conventional charges $(\mathbb 1,\psi,\sigma)$:

using TensorCategories, Oscar
C = ising_category()
vacuum, psi, sigma = simples(C)
@assert vacuum == one(C)
@assert psi ⊗ psi == vacuum
@assert psi ⊗ sigma == sigma ⊗ psi == sigma
@assert sigma ⊗ sigma == vacuum ⊕ psi
@assert dim(sigma)^2 == 2
@assert dim(C) == 4
dim.(simples(C))

3-element Vector{AbsSimpleNumFieldElem}:
 1
 1
 √2

Thus the package's global dimension is

\[\label{eq:physics-ising-global-dimension} \dim(\mathcal C)=\sum_a d_a^2=4.\]

Physicists usually call $\mathcal D=2$ the total quantum dimension. The method dim(C) returns $\mathcal D^2$, whereas dim(sigma) returns the quantum dimension $d_\sigma$.

Fusion spaces are not object dimensions

For three Ising anyons with total charge $\sigma$,

\[\label{eq:physics-ising-four-anyon-space} \mathcal H^{\sigma}_{\sigma\sigma\sigma} =\operatorname{Hom}((\sigma\otimes\sigma)\otimes\sigma,\sigma)\]

has dimension two. Its usual fusion-tree basis is indexed by the intermediate charges $\mathbb 1$ and $\psi$:

H = Hom((sigma ⊗ sigma) ⊗ sigma, sigma)
@assert int_dim(H) == 2
@assert size(matrix(id(sigma))) == (1,1)
int_dim(H)

2

The $1\times1$ matrix representing id(sigma) does not say that $d_\sigma=1$. This matrix records the multiplicity coordinates of the identity morphism in the package's split semisimple realization; quantum dimension is categorical trace data. The distinction is developed under Fiber functors and semisimple coordinates.

$F$- and $R$-symbols

An $F$-matrix represents the associator between two fusion-tree bases. In the package's projection convention, an $R$-matrix represents the pullback by the braiding

\[\label{eq:physics-braiding-pullback} c_{a,b}^{*}:V_{ba}^{c}\longrightarrow V_{ab}^{c}, \qquad p\longmapsto p\circ c_{a,b}.\]

Equivalently, the braiding itself maps the composition-dual splitting space $\operatorname{Hom}(c,a\otimes b)$ to $\operatorname{Hom}(c,b\otimes a)$. Consequently, the entries of $F$- and $R$-matrices depend on the chosen projection and splitting bases, on the direction of the structural maps, and on the ordering of multiplicity indices. They are not determined by the fusion rules alone.

TensorCategories.jl first introduces the underlying skeletal fusion model, where the abstract model is followed by the precise symbol conventions. This order is essential when translating formulas or data from the physics literature.

$S$, $T$, topological spins, and CFT conventions

The formal hypotheses and definitions appear under Premodular and modular categories. In those hypotheses, smatrix(C) returns the unnormalized pivotal trace of double braiding, while the package uses

\[\label{eq:physics-modular-data-normalization} S^{\mathrm{norm}}=\frac{1}{\sqrt{\dim(\mathcal C)}}S, \qquad T^{\mathrm{cat}}=\operatorname{diag}(\theta_a).\]

These are the categorical twist and $S$-matrix conventions of Etingof et al. (2015), §§8.10 and 8.13, especially Eq. (8.46).

The square root in normalized_smatrix(C) is a choice over a general coefficient field. In a unitary realization one chooses the positive total quantum dimension.

For RCFT characters, a common convention is

\[\label{eq:physics-rcft-t-matrix} T^{\mathrm{RCFT}}_{aa} =\exp\!\left(2\pi i\left(h_a-\frac{c}{24}\right)\right) =e^{-2\pi i c/24}\theta_a, \qquad \theta_a=e^{2\pi i h_a}.\]

Thus tmatrix(C) is the categorical twist matrix, without the overall central-charge phase. A unitary modular category constrains the topological central charge through its modular data, but it does not determine a full RCFT central charge or conformal weights beyond their categorical phase data. The relation between RCFT fusing and braiding data and categorical modular data is described by Moore and Seiberg (1989).

Exact and numerical use

Fusion data may be stored exactly over a number field or evaluated over complex balls at a chosen working precision. Exact coefficients can have several complex embeddings, and different embeddings can change unitarity or phases. The coefficient-field chapter explains these choices. The numerical-computation chapter explains ball arithmetic, and the later numerical fusion-category chapter applies it to coherence, unitarity, and modularity. Numerical center computations are introduced in the Drinfeld-center chapter, after half-braidings have been defined.

Scope and current limitations

A modular tensor category records topological and chiral categorical data; it is not by itself a complete two-dimensional CFT. TensorCategories.jl does not construct characters, position-dependent conformal blocks, operator-product-expansion (OPE) coefficients, or modular-invariant partition functions. Full RCFT constructions require additional data; for example, Fuchs et al. (2002) use a symmetric special Frobenius algebra in a modular tensor category.

The package supplies associators, braidings, and local $F$- and $R$-symbols, from which braid actions can be assembled. It currently has no public high-level API that accepts a braid word and returns its matrix in a chosen multi-anyon fusion-tree basis. The ordinary braiding(X,Y) method is a categorical structural map, not such a braid-word interface.

Continue with Models and the category interface. Readers interested primarily in fusion data can later follow skeletal fusion categories, precise symbol conventions, and numerical fusion categories, in that order.