\(\text{Fib}:\ \text{FR}^{2,0}_{2}\)
Fusion Rules
\[\begin{array}{|ll|} \hline \mathbf{1} & \mathbf{2} \\ \mathbf{2} & \mathbf{1}+\mathbf{2} \\ \hline \end{array}\]Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.61803\) | \(\phi\) |
| \(\mathcal{D}_{FP}^2\) | \(3.61803\) | \(2 + \phi\) |
The Frobenius-Perron dimension of the non-trivial element equals the golden ratio \(\phi = \frac{1+\sqrt{5}}{2}\) and appears in a lot of the formulas. To save space we will keep using the abbreviation \(\phi\) in what follows.
Characters
The symbolic character table is the following
\[\begin{array}{|cc|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1 & \phi \\ 1 & -\phi^{-1} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rr|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1.000 & 1.618 \\ 1.000 & -0.6180 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{2 + \phi}}\left(\begin{array}{cc} 1 & \phi \\ \phi & -1 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{2}{5}\right) \\\left(0,\frac{2}{5}\right)\end{array}\) |
Adjoint Subring
The adjoint subring is the ring itself.
The upper central series is trivial.
Universal grading
This fusion ring allows only the trivial grading.
Categorifications
There are two Galois conjugate solutions to the pentagon equations, one unitary, one not unitary. Each of these gives rise to a mirror pair of hexagon solutions. Both pairs are modular and the pair corresponding to the unitary pentagon solution is unitary. We give a table for one representative of each pair. The quantities for the mirror theory are also easily obtained from these tables; quantum dimensions and Frobenius-Schur indicators are invariant, central charge and weights get a minus sign and the S-matrix must be complex conjugated.
\[\begin{array}{|l|c|} \hline c=\frac{14}{5} & \mathcal{D}=\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} ~~~~~\mathbf{(G_2)_{1}} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & \phi \\ h & 0 & \frac{2}{5} \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & \phi \\ \phi & -1 \end{array} \right) \\~&~\\\hline \end{array}\] \[\begin{array}{|l|c|} \hline c=-\frac{2}{5} & \mathcal{D}^2=\frac{1}{2} \left(5-\sqrt{5}\right) ~~~~~\mathbf{\textstyle Yang-Lee} \\ \hline ~&~ \\ \begin{array}{|l|rr|} \hline & \psi_0 & \psi_1 \\ \hline d & 1 & -\phi^{-1} \\ h & 0 & \frac{4}{5} \\ \kappa & 1 & 1 \\ \hline \end{array} & \mathcal{D} S=\left( \begin{array}{rr} 1 & -\phi^{-1} \\ -\phi^{-1} & -1 \end{array} \right) \\~&~\\\hline \end{array}\]Data
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