$\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}$

Fusion Rules

$\begin{array}{|ll|} \hline \mathbf{1} & \mathbf{2} \\ \mathbf{2} & \mathbf{1} \\ \hline \end{array}$

Frobenius-Perron Dimensions

Particle	Numeric	Symbolic
$\mathbf{1}$	$1.$	$1$
$\mathbf{2}$	$1.$	$1$
$\mathcal{D}_{FP}^2$	$2.$	$2$

Characters

The symbolic character table is the following

$\begin{array}{|cc|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1 & 1 \\ 1 & -1 \\ \hline \end{array}$

The numeric character table is the following

$\begin{array}{|rr|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1.000 & 1.000 \\ 1.000 & -1.000 \\ \hline \end{array}$

Representations of $SL_2(\mathbb{Z})$

The matching $S$-matrices and twist factors are the following

$S$-matrix	Twist factors
$\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \\\end{array}\right)$	$\begin{array}{l}\left(0,-\frac{1}{4}\right) \\\left(0,\frac{1}{4}\right)\end{array}$

Adjoint Subring

The adjoint subring is the trivial ring.

The upper central series is the following: $\mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}$

Universal grading

Each particle can be graded as follows: $\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}'$, where the degrees form the group $\mathbb{Z}_2$ with multiplication table:

$\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}$

Categorifications

For this section we will use the the following labels for the elements: $\psi_0 = \mathbf{1}, \psi_1 = \mathbf{2}$.

There are 2 solutions to the pentagon equations, each of which gives rise to 2 solutions to the hexagon equations, so 4 solutions in total. For each of these solutions, we can still choose the quantum dimension $d_1$ of the nontrivial element type to be equal to either $1$ or $-1$. Choosing $d_1=1$ results in four unitary anyon models.

We have either two bosons

$\begin{array}{|l|l|} \hline {\rm not~modular} & \mathcal{D}=\sqrt{2} ~~~~~ \mathbf{\mathbb{Z}_{2}^{(0)}} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & 1 \\ h & 0 & 0 \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right) \\~&~\\\hline \end{array}$

or a boson and a fermion

$\begin{array}{|r|r|} \hline {\rm not~modular} & \mathcal{D}=\sqrt{2} ~~~~~ \mathbf{\mathbb{Z}_{2}^{(1)}} \\ \hline ~&~ \\ \begin{array}{|r|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & 1 \\ h & 0 & \frac{1}{2} \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & 1 \\ 1 & 1 \end{array} \right) \\~&~\\\hline \end{array}$

or a boson and a semion, which can occur with two chiralities

$\begin{array}{|l|l|} \hline c=1 & \mathcal{D}=\sqrt{2} ~~~~~ \mathbf{SU(2)_{1}} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & 1 \\ h & 0 & \frac{1}{4} \\ \kappa & 1 & -1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right) \\~&~\\\hline \hline c=-1 & \mathcal{D}=\sqrt{2} ~~~~~ \mathbf{SU(2)_{1}^{\%}} \\ \hline ~&~ \\ \begin{array}{|r|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & 1 \\ h & 0 & -\frac{1}{4} \\ \kappa & 1 & -1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array} \right) \\~&~\\\hline \end{array}$

Only the semionic theories are modular.

If we choose $d_1=-1$, we get a further four models, which are not unitary. As in the unitary case, only the semionic theories are modular and their data are given below. The different choice of $d_1$ affects the S-matrix and the Frobenius-Schur indicator $\kappa_1$, but the spin factors are unchanged. Note that the central charge for a non-unitary modular theory should be read modulo 4, rather than modulo 8 since in this case it is not clear which root of $\mathcal{D}^2$ one should take in the expression for $c$.

$\begin{array}{|l|l|} \hline c=1 & \mathcal{D}=\sqrt{2} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & -1 \\ h & 0 & \frac{1}{4} \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & -1 \\ -1 & -1 \end{array} \right) \\~&~\\\hline \hline c=-1 & \mathcal{D}=\sqrt{2} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & -1 \\ h & 0 & -\frac{1}{4} \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & -1 \\ -1 & -1 \end{array} \right) \\~&~\\\hline \end{array}$

Data

Download links for numeric data:

• Multiplication Table
• Quantum Dimensions
• Character Table
• S-Matrix 1 , Twist Factors1