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