\(\text{Ising}:\ \text{FR}^{3,0}_{1}\)
Fusion Rules
\[\begin{array}{|lll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathcal{D}_{FP}^2\) | \(4.\) | \(4\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1 & 1 & \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ 1 & -1 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline 1.000 & 1.000 & 1.414 \\ 1.000 & 1.000 & -1.414 \\ 1.000 & -1.000 & 0 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{2}\left(\begin{array}{ccc} 1 & 1 & \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{2},\frac{1}{16}\right) \\\left(0,\frac{1}{2},\frac{3}{16}\right) \\\left(0,\frac{1}{2},\frac{5}{16}\right) \\\left(0,\frac{1}{2},\frac{7}{16}\right) \\\left(0,\frac{1}{2},-\frac{7}{16}\right) \\\left(0,\frac{1}{2},-\frac{5}{16}\right) \\\left(0,\frac{1}{2},-\frac{3}{16}\right) \\\left(0,\frac{1}{2},-\frac{1}{16}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}\), form the adjoint subring \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) .
The upper central series is the following: \(\text{Ising} \underset{ \mathbf{1}, \mathbf{2} }{\supset} \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}'\), 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
There are 8 unitary modular realizations, distinguished by central charge, realized as \(SO(2k+1)_1\) for \(1\leq k\leq 8\). There are also 8 non-unitary obtained by changing the spherical structure. DGNO show that any fusion category with the Ising fusion rules admits a braided structure, and is automatically non-degenerate (i.e. modular for any choice of spherical structure).
Data
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