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
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
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
where Nab
= (a b)(c). When no
confusion arises, we write a b simply as ab, so
= a a. Then the
conditions on are: for all a, b, c, d L,
(i) (a b) c = a (b c), i.e.,

NabNxc x d =

(ii) Na1 c = N1a c = δca
(iii) Nab 1 = Nba 1 = δbˆ
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,
for g, h G. When G = Z2, this is the Ising fusion rule.
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