Fib: FR22,0\text{Fib}:\ \text{FR}^{2,0}_{2}

Fusion Rules

1221+2\begin{array}{|ll|} \hline \mathbf{1} & \mathbf{2} \\ \mathbf{2} & \mathbf{1}+\mathbf{2} \\ \hline \end{array}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.618031.61803 ϕ\phi
DFP2\mathcal{D}_{FP}^2 3.618033.61803 2+ϕ2 + \phi

The Frobenius-Perron dimension of the non-trivial element equals the golden ratio ϕ=1+52\phi = \frac{1+\sqrt{5}}{2} and appears in a lot of the formulas. To save space we will keep using the abbreviation ϕ\phi in what follows.

Characters

The symbolic character table is the following

121ϕ1ϕ1\begin{array}{|cc|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1 & \phi \\ 1 & -\phi^{-1} \\ \hline \end{array}

The numeric character table is the following

121.0001.6181.0000.6180\begin{array}{|rr|} \hline \mathbf{1} & \mathbf{2} \\ \hline 1.000 & 1.618 \\ 1.000 & -0.6180 \\ \hline \end{array}

Representations of SL2(Z)SL_2(\mathbb{Z})

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
12+ϕ(1ϕϕ1)\frac{1}{\sqrt{2 + \phi}}\left(\begin{array}{cc} 1 & \phi \\ \phi & -1 \\\end{array}\right) (0,25)(0,25)\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.

c=145D=12(5+5)     (G2)1  ψ0ψ1d1ϕh025κ11DS=(1ϕϕ1)  \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} c=25D2=12(55)     YangLee  ψ0ψ1d1ϕ1h045κ11DS=(1ϕ1ϕ11)  \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

Download links for numeric data: