$so(2p+1)_2$

These fusion rings are called metaplectic fusion rings. We assume that $p\geq 1$.

Particle labels

The fusion ring has the following basis \[ ( \mathbf{1}, \epsilon, \psi_+, \psi_-, \phi_1, \ldots, \phi_{p}), \] of $p + 4$ elements. The particles have the following quantum dimensions

Particle	$\fpdim$
$\mathbf{1}$	$1$
$\epsilon $	$1$
$\psi_\pm $	$\sqrt{2p+1}$
$\phi_j $	$2$

Thus we have $\mathcal{D}_{FP}^2 = 4(2p+1)$.

Fusion Rules

The fusion rules read as follows.

The action of $\epsilon$ is

\[\begin{array}{l} \epsilon \times \epsilon = \mathbf{1} \\ \epsilon \times \psi_\pm =\psi_\mp \\ \epsilon \times \phi_j = \phi_j \end{array}\]

The action of $\psi_\pm$ is

\[\begin{array}{l} \psi_\pm \times \psi_\pm = \mathbf{1} + \sum_j \phi_j \\ \psi_\pm \times \psi_\mp = \epsilon + \sum_j \phi_j \\ \psi_\pm \times \phi_j = \psi_+ + \psi_- \end{array}\]

Finally, the $\phi_i$ fuse amongst themselves as

\[\begin{array}{l} \phi_j \times \phi_j = \begin{cases} \mathbf{1} + \epsilon + \phi_{2j} & \text{if $2j \leq p$}\\ \mathbf{1} + \epsilon + \phi_{2p-2j} & \text{if $2j > p$}\\ \end{cases} \\ \phi_j \times \phi_{k \neq j} = \begin{cases} \phi_{|j-k|} + \phi_{j+k} & \text{if $j+k \leq p$}\\ \phi_{|j-k|} + \phi_{2p-j-k} & \text{if $j+k > p$} \end{cases} \end{array}\]

Modular Data

The solutions of the pentagon equations are labelled by the tuple $(p,r,\kappa)$, where $r$ is an odd integer $1\leq r < 2p+1$ such that $\gcd(r,2p+1)=1$ and $\kappa = \pm 1$. For each solution, there are two spherical structures; for one of them, all quantum dimensions are positive. Below, we only consider those cases.

The solutions of the hexagon equations are labelled by the tuple $(p,r,\kappa,\lambda)$, where $\lambda = \pm 1$. Swapping $\lambda = +1 \leftrightarrow \lambda = -1$ merely inverts the $R$-symbols.

$S$-matrix

To specify the modular $S$-matrix, we state the elements of $\tilde{S} = 2\sqrt{2p+1} S$.

\[\begin{array}{llll} \tilde{S}_{\mathbf{1},\mathbf{1}} = 1 \\ \tilde{S}_{\mathbf{1},\epsilon} = 1 & \tilde{S}_{\epsilon,\epsilon} = 1\\ \tilde{S}_{\mathbf{1},\psi_\pm} = \sqrt{2p+1} & \tilde{S}_{\epsilon,\psi_\pm} = -\sqrt{2p+1} & \tilde{S}_{\psi_\pm,\psi_\pm} = -\kappa \sqrt{2p+1} (2|2p+1) \\ & & \tilde{S}_{\psi_\pm,\psi_\mp} = \kappa \sqrt{2p+1} (2|2p+1)\\ \tilde{S}_{\mathbf{1},\phi_j} = 2 & \tilde{S}_{\epsilon,\phi_j} = 2 & \tilde{S}_{\psi_\pm,\phi_j} = 0 & \tilde{S}_{\phi_j,\phi_k} = 4 \cos\bigl(\frac{2 \pi jkr}{2p+1}\bigr) \end{array}\]

Here, $(2|2p+1)$ denotes the Jacobi symbol.

Twist factors

The twist factors are given by

\[\begin{array}{l} \theta_{\mathbf{1}} = 1 \\ \theta_{\epsilon} = 1 \\ \theta_{\psi_\pm} = \mp (-1)^{\frac{\kappa \lambda r}{2}} e^{\frac{i \pi \lambda r}{4}( (2p+1|r) + \kappa(2|2p+1) -p)} \\ \theta_{\phi_j} = (-1)^{j} e^{i \pi \frac{\lambda r j^2}{2p+1}} \end{array}\]

Central charge

The topological central charge is given by $c_{\rm top} = 2p(\lambda+2p)−2+2(r|2p+1) \bmod 8$.

$so(2p+1)_2$ CFT

The $so(2p+1)_2$ CFTs correspond to the following values, $r=1$, $\kappa = (2p+1|2) = (-1)^{p(p+1)/2}$ and $\lambda = -1$. These CFTs have central charge $c=2p$, in agreement with the fomula above.

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\(so(2p+1)_2\)