Z 2 : FR 1 2 , 0 \mathbb{Z}_2:\ \text{FR}^{2,0}_{1} Z 2 : FR 1 2 , 0
Fusion Rules
1 2 2 1 \begin{array}{|ll|}
\hline
\mathbf{1} & \mathbf{2} \\
\mathbf{2} & \mathbf{1} \\
\hline
\end{array} 1 2 2 1
Frobenius-Perron Dimensions
Particle
Numeric
Symbolic
1 \mathbf{1} 1
1. 1. 1 .
1 1 1
2 \mathbf{2} 2
1. 1. 1 .
1 1 1
D F P 2 \mathcal{D}_{FP}^2 D F P 2
2. 2. 2 .
2 2 2
Characters
The symbolic character table is the following
1 2 1 1 1 − 1 \begin{array}{|cc|}
\hline
\mathbf{1} & \mathbf{2} \\
\hline
1 & 1 \\
1 & -1 \\
\hline
\end{array} 1 1 1 2 1 − 1
The numeric character table is the following
1 2 1.000 1.000 1.000 − 1.000 \begin{array}{|rr|}
\hline
\mathbf{1} & \mathbf{2} \\
\hline
1.000 & 1.000 \\
1.000 & -1.000 \\
\hline
\end{array} 1 1 . 0 0 0 1 . 0 0 0 2 1 . 0 0 0 − 1 . 0 0 0
Representations of S L 2 ( Z ) SL_2(\mathbb{Z}) S L 2 ( Z )
The matching S S S -matrices and twist factors are the following
S S S -matrix
Twist factors
1 2 ( 1 1 1 − 1 ) \frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \\\end{array}\right) 2 1 ( 1 1 1 − 1 )
( 0 , − 1 4 ) ( 0 , 1 4 ) \begin{array}{l}\left(0,-\frac{1}{4}\right) \\\left(0,\frac{1}{4}\right)\end{array} ( 0 , − 4 1 ) ( 0 , 4 1 )
Adjoint Subring
The adjoint subring is the trivial ring.
The upper central series is the following:
Z 2 ⊃ 1 Trivial \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial} Z 2 1 ⊃ Trivial
Universal grading
Each particle can be graded as follows: deg ( 1 ) = 1 ′ , deg ( 2 ) = 2 ′ \text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}' deg ( 1 ) = 1 ′ , deg ( 2 ) = 2 ′ , where the degrees form the group Z 2 \mathbb{Z}_2 Z 2 with multiplication table:
1 ′ 2 ′ 2 ′ 1 ′ \begin{array}{|ll|}
\hline
\mathbf{1}' & \mathbf{2}' \\
\mathbf{2}' & \mathbf{1}' \\
\hline
\end{array} 1 ′ 2 ′ 2 ′ 1 ′
Categorifications
For this section we will use the the following labels for the elements: ψ 0 = 1 , ψ 1 = 2 \psi_0 = \mathbf{1}, \psi_1 = \mathbf{2} ψ 0 = 1 , ψ 1 = 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 d_1 d 1 of the nontrivial element type to be equal to either 1 1 1 or − 1 -1 − 1 . Choosing d 1 = 1 d_1=1 d 1 = 1 results in four unitary anyon models.
We have either two bosons
n o t m o d u l a r D = 2 Z 2 ( 0 ) ψ 0 ψ 1 d 1 1 h 0 0 κ 1 1 D S = ( 1 1 1 1 ) \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} n o t m o d u l a r d h κ ψ 0 1 0 1 ψ 1 1 0 1 D = 2 Z 2 ( 0 ) D S = ( 1 1 1 1 )
or a boson and a fermion
n o t m o d u l a r D = 2 Z 2 ( 1 ) ψ 0 ψ 1 d 1 1 h 0 1 2 κ 1 1 D S = ( 1 1 1 1 ) \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} n o t m o d u l a r d h κ ψ 0 1 0 1 ψ 1 1 2 1 1 D = 2 Z 2 ( 1 ) D S = ( 1 1 1 1 )
or a boson and a semion, which can occur with two chiralities
c = 1 D = 2 S U ( 2 ) 1 ψ 0 ψ 1 d 1 1 h 0 1 4 κ 1 − 1 D S = ( 1 1 1 − 1 ) c = − 1 D = 2 S U ( 2 ) 1 % ψ 0 ψ 1 d 1 1 h 0 − 1 4 κ 1 − 1 D S = ( 1 1 1 − 1 ) \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} c = 1 d h κ ψ 0 1 0 1 ψ 1 1 4 1 − 1 c = − 1 d h κ ψ 0 1 0 1 ψ 1 1 − 4 1 − 1 D = 2 S U ( 2 ) 1 D S = ( 1 1 1 − 1 ) D = 2 S U ( 2 ) 1 % D S = ( 1 1 1 − 1 )
Only the semionic theories are modular.
If we choose d 1 = − 1 d_1=-1 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 d_1 d 1 affects the S-matrix and the Frobenius-Schur indicator κ 1 \kappa_1 κ 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 D 2 \mathcal{D}^2 D 2 one should take in the expression for c c c .
c = 1 D = 2 ψ 0 ψ 1 d 1 − 1 h 0 1 4 κ 1 1 D S = ( 1 − 1 − 1 − 1 ) c = − 1 D = 2 ψ 0 ψ 1 d 1 − 1 h 0 − 1 4 κ 1 1 D S = ( 1 − 1 − 1 − 1 ) \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} c = 1 d h κ ψ 0 1 0 1 ψ 1 − 1 4 1 1 c = − 1 d h κ ψ 0 1 0 1 ψ 1 − 1 − 4 1 1 D = 2 D S = ( 1 − 1 − 1 − 1 ) D = 2 D S = ( 1 − 1 − 1 − 1 )
Data
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