Braided monoidal category

A monoidal category with natural invertible braidings

c_{X, Y} : X \otimes Y \to Y \otimes X

satisfying the two hexagon axioms. The equation

c_{Y, X} c_{X, Y} = 1

is an additional symmetric condition, not a braided axiom.

Modular tensor category

A finite semisimple ribbon braided monoidal category with nondegenerate modular S-matrix. For a unitary anyon theory, this means that the only transparent anyon is the vacuum anyon. Such data define a three-dimensional topological quantum field theory.

Chiral central charge ( $c (mod 8)$ )

The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through

e^{2 πi c /8} = D^{- 1} \sum_{a} d_{a}^{2} θ_{a}

. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

Handle-cut formula for anyon ground-state degeneracy ( $dim V (Σ_{g}) = [(\sum_{a} N_{a} N_{\overset{a}{ˉ}})^{g}]_{11}$ )

Cutting one nonseparating circle on each handle of a closed orientable surface gives a sphere with

g

pairs of oppositely oriented boundary circles. The topological quantum field theory gluing rule sums over the paired anyon labels

a, \overset{a}{ˉ}

. The resulting anyon fusion matrix formula becomes

dim V (Σ_{g}) = D^{2 g - 2} \sum_{a} d_{a}^{2 - 2 g}

by the Verlinde formula.

Modular S-matrix ( $S_{ab}$ )

The matrix of Hopf link evaluations divided by the total quantum dimension

D

. In a unitary modular tensor category it is a symmetric unitary matrix, with

S_{1 a} = d_{a} / D

and

S_{\overset{a}{ˉ} b} = S_{ab}^{*}

. Its columns diagonalize the anyon fusion matrices through the Verlinde formula.

Verlinde formula ( $N_{ab}^{c} = \sum_{x} S_{a x} S_{b x} S_{c x}^{*} / S_{1 x}$ )

The simultaneous diagonalization formula for the anyon fusion matrices of a modular tensor category. The eigenvalue of

N_{b}

in column

x

of the modular S-matrix is

S_{b x} / S_{1 x}

.

Free braided monoidal category on one object ( $F b r$ )

The braided monoidal category generated by one object: objects are parenthesized tensor expressions in that object and the unit, and morphisms are structural isomorphisms and crossings subject to precisely the monoidal category and braiding axioms.

Braid category ( $B$ )

The braided monoidal category with objects

n \geq 0

, endomorphism groups

B_{n}

, no arrows between unequal objects, tensor given by juxtaposition, and block-crossing braiding. Its tensor is strict.

Braided monoidal functor

A monoidal functor between braided monoidal categories compatible with their braidings:

F (c_{X, Y}) F_{2} (X, Y) = F_{2} (Y, X) c_{FX, F Y}

.

Braiding ( $c_{X, Y}$ )

The natural tensor interchange isomorphism in a braided monoidal category, constrained by the two hexagon axioms.

Hexagon identity for braiding

The compatibility equations between braiding and the associator in a braided monoidal category. Omitting the associators from the notation, they read

c_{a, b \otimes c} = (1_{b} \otimes c_{a, c}) (c_{a, b} \otimes 1_{c})

and

c_{a \otimes b, c} = (c_{a, c} \otimes 1_{b}) (1_{a} \otimes c_{b, c})

. Thus exchanging a composite with another object is the sequence of exchanges of its factors.

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Braided monoidal category by

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A *braided monoidal category* is a particular type of category that combines the structure of a monoidal category with a braiding. To understand this structure, let's unpack a few key concepts. 1. **Monoidal Category**: A monoidal category consists of: - A category \( C \). - A tensor product (a bifunctor) \( \otimes: C \times C \to C \).

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