This post is jointly written by
Vaibhav Kalvakota and Aayush Verma (and note that neither of us is an expert in anyons!). We discuss the anyon statistics and later relate the ideas of modular tensor categories with them.
We begin with a couple of words about anyon statistics. In three dimensions, the exchange of two indistinguishable particle positions $(r,p)$ makes the wavefunction $\psi(r,p)$ acquire a phase, but when we exchange the positions once more, it is equivalent to a trivial loop. Hence, the phase acquired must satisfy some constraints $$\psi(r,p) = e^{2i\theta} \psi(r,p)$$ where $\theta=n\pi$, which are exactly the constraints that define boson exchange statistics (when $n$ is an even integer) or fermion exchange statistics (when $n$ is an odd integer).
However, in two dimensions, a closed loop exchange is not equivalent to a trivial loop. The $\theta$-statistics, which are followed by neither bosons nor fermions, lead to anyon statistics in two dimensions. And since the following is true
the particles are defined using a braid group (which is the fundamental group of the configuration space) and not a permutation group. Moreover, in anyon exchange, the history of double exchanges is not forgotten, unlike for bosons and fermions, where it is a trivial exchange, as we remarked earlier. For more on anyon statistics, see
this paper by Rao.
A fusion category $\mathcal{C}$ is a
semisimple tensor category which is rigid, composed of finite
isoclasses of simples, and has a simple unit object (refer to
this for more). The most important algebraic
rules in $\mathcal{C}$ is the fusion rules; given $V_{i}$ and $V_{j}$
simples, their tensor product (fusion) can be decomposed into some other
simple $V_k$ as \[ V_{i}\boxtimes V_{j} = \bigoplus _{k\in I} N _{ij}^{k}V_{k}\;, \] where
$I$ indexes the objects in $\mathcal{C}$, and $N_{ij}^{k}$ are
non-negative integers. Physically, $N _{ij}^{k}$ accounts for the number
of distinct topologically invariant ways in which anyons $i, j$ fuse to
create anyon $k$. Fusion of two anyons does not give a unique anyon, as we saw that there could be more ways to do the fusion. Fusion rules are associative. These fusion rules create a fusion ring, which is
also the Grothendieck group of the category $\mathcal{C}$. When there are more than two anyons, the change of basis in the state space (fusion in a particular order) is given by F-matrices (see
this talk and
this paper). Fusion of non-abelian anyons is more complicated than that of abelian anyons, see
this.
Anyon exchanges are essentially just braiding across objects like $\mathbf{b}_{ij}: V_{i} \boxtimes V_{j} \rightarrow V_{j} \boxtimes V_i$. Here, the hexagonal identities apply as usual:
For
each object, we also attribute a twist $\theta _{i}$, and for anyons, for
a full $2\pi $ rotation, we obtain a phase or a topological spin. Not
that this only happens in two dimensions, and in ordinary 3d, this would
just be the usual particle spin statistics. Since we now have two pieces
of data corresponding to the $S$ and $T$ matrices -- for the braiding
and twisting respectively, they generate a projective representation
of SL$_2(Z)$, from which we also get the charge conjugation matrix and
the central charge; the $S$ and $T$ matrices follow: $$(ST)^3=\Lambda C, \\ S^2=C,\\ C^2 = I_n, $$ where $\Lambda $ is \[ \Lambda = \frac{1}{\mathcal{D}} \sum d_{i}^2\theta _{i} = e^{2\pi i c/8}\;. \] Here,
$d_{i}$ is the quantum dimension and $\mathcal{D}$ is the global
quantum dimension $\sqrt{\sum _{i} d_{i}^2}$. Moreover, the Verlinde formula describes the fusion coefficients in terms of the $S$-matrix \[ N_{ij}^{k} = \sum \frac{S_{ax}S_{bx}S^*_{cx}}{S_{0a}}\;, \] where $S_{0a}$ is just $d_a/\mathcal{D}$ which is an important result in CFT as well as modular tensor categories.
This way, not only do we
obtain a fusion category, but something with braiding structure and
modular data (i.e., the $S$-matrix is invertible), which leads us to
modular tensor categories more broadly. These have important consequences in finding a consistent theory of anyons.
