CHAPTER 4

Ribbon Fusion Categories

This chapter introduces the most important concept of the book: ribbon fusion

categories (RFCs). In the literature, they are also called premodular categories.

With an extra nondegeneracy condition for the braiding, they are called modular

tensor categories (MTCs). Unitary modular tensor categories (UMTCs) are the

algebraic models of anyons and the algebraic data for unitary TQFTs. In the end

we list all UMTCs of rank ≤ 4. All our linear categories are over C.

4.1. Fusion rules and fusion categories

Group theory is an abstraction of symmetry, which is fundamental to mathe-

matics and physics. Finite groups are closely related to the classification of crystals.

Fusion categories can be regarded as quantum generalizations of finite groups. The

simplest finite groups are abelian. Pursuing this analogy, we can consider RFCs as

quantum generalizations of finite abelian groups.

Definition 4.1.

(1) A label set L is a finite set with a distinguished element 1 and an involution

ˆ: L → L such that

ˆ

1 = 1. Elements of L are called labels, 1 is called the

trivial label, sometimes written 0, and ˆ is called duality.

(2) A fusion rule on a label set L is a binary operation ⊗: L × L →

NL,

where

NL

is the set of all maps from L to N = {0, 1, 2,...} satisfying the

following conditions. First we introduce some notation. Given a, b ∈ L,

we will write formally a ⊗ b = Nabc

c

where Nab

c

= (a ⊗ b)(c). When no

confusion arises, we write a ⊗ b simply as ab, so

a2

= a ⊗ a. Then the

conditions on ⊗ are: for all a, b, c, d ∈ L,

(i) (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c), i.e.,

∑

x∈L

NabNxc x d =

∑

x∈L

NbcNaxdx

(ii) Na1 c = N1a c = δca

(iii) Nab 1 = Nba 1 = δbˆ

a

We say a triple of labels (a, b, c) is admissible if Nab c = 0. We often refer to an

instance of the equation a⊗b = Nabc c as a fusion rule, though technically ⊗ itself

is the fusion rule. Since 1 ⊗ a = a = a ⊗ 1, in the future we would not list such

trivial fusion rules.

Example 4.2. A finite group G is a label set with elements of G as labels, trivial

label 1, and ˆ g = g−1. A fusion rule on G is g ⊗ h = gh, i.e., (g ⊗ h)(k) = δgh,k.

Example 4.3 (Tambara-Yamagami [TY]). Given a finite group G, the label

set L = G {m}, where m / ∈ G, with fusion rule

g ⊗ h = gh, m ⊗ g = g ⊗ m = m,

m2

=

g∈G

g

for g, h ∈ G. When G = Z2, this is the Ising fusion rule.

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http://dx.doi.org/10.1090/cbms/112/04