$\mathbb{Z}_n$

Particle labels

The particles are labelelled $\phi_i$ with $i=0,1,2,\ldots, n-1$.

Fusion Rules

\[\begin{array}{|l|} \hline \phi_i \times \phi_j = \phi_{i+j\bmod n} \\ \hline \end{array}\]

Quantum Dimensions

Particle	Numeric	Symbolic
$\phi_i$	$1.$	$1$
$\mathcal{D}_{FP}^2$		$n$

Modular Data

We provide two sets of solutions for the modular data.

Case 1

For case 1, the $S$-matrices and twist factors depend on the parameter $p = 0, 1, \ldots n-1$. These theories are modular, provided that $n$ is odd, and $\gcd(p,n)=1$. Thus, the central charge is only defined under these conditions.

$S$-matrix elements	Twist factors
$S_{j,k} = \frac{1}{\sqrt{n}}e^{2 \pi i \frac{2p j k}{n}}$	$\theta_j = e^{2 \pi i \frac{p j^2}{n}}$

The topological central charge, which is defined modulo 8, is given by

\[\begin{array}{l} c_{\rm top} = 0 \quad \text{for $n=1 \bmod4 \wedge (p|n) = 1$} \\ c_{\rm top} = 2 \quad \text{for $n=3 \bmod4 \wedge (p|n) = 1$} \\ c_{\rm top} = 4 \quad \text{for $n=1 \bmod4 \wedge (p|n) = -1$} \\ c_{\rm top} = 6 \quad \text{for $n=3 \bmod4 \wedge (p|n) = -1$} \end{array}\]

Here, $(a|n)$ denotes the Jacobi symbol, which arises in the study of quadratic Gauss sums, which naturally appear when calculating the central charge modulo 8.

We note that the cases with $n$ odd and $p = (n-1)/2$ correspond to $su(n)_1$, which has central charge $c = n-1$.

Case 2

For case 2, the $S$-matrices and twist factors again depend on $p = 0, 1, \ldots n-1$. These theories are modular, provided that $\gcd(2p+1,n)=1$, in which case the central charge is defined.

$S$-matrix elements	Twist factors
$S_{j,k} = \frac{1}{\sqrt{n}}e^{2 \pi i \frac{(2p+1) j k}{2n}}$	$\theta_j = e^{2 \pi i \bigl(\frac{(2p+1) j^2}{2n} + \frac{n j}{2}\bigr)}$

For $n$ odd, the topological central charge is given by

\[\begin{array}{l} c_{\rm top} = (1-n) \bmod 8 \quad \text{for $(2p+1|n) = 1$} \\ c_{\rm top} = (5-n) \bmod 8 \quad \text{for $(2p+1|n) = 1$} \end{array}\]

For $n$ even, the topological central charge is given by

\[\begin{array}{l} c_{\rm top} = 1 \quad \text{for $2p+1 = 1 \bmod 4 \wedge (2n|2p+1) = 1$} \\ c_{\rm top} = 3 \quad \text{for $2p+1 = 3 \bmod 4 \wedge (2n|2p+1) = -1$} \\ c_{\rm top} = 5 \quad \text{for $2p+1 = 1 \bmod 4 \wedge (2n|2p+1) = -1$} \\ c_{\rm top} = 7 \quad \text{for $2p+1 = 3 \bmod 4 \wedge (2n|2p+1) = 1$} \ . \end{array}\]

Again, $(a|n)$ denotes the Jacobi symbol.

The cases with $n$ even and $p = \frac{n}{2}-1$ correspond to $su(n)_1$, which has central charge $c = n-1$.

The cases with $n$ even and $p=1$ correspond to $u(1)_n$ (that is, the compactified boson cft with $n$ primary fields), which has central charge $c=1$.

NOTE: I need to check if $n$ is allowed to be odd in case 2 !

Particle	Numeric	Symbolic
\(\phi_i\)	\(1.\)	\(1\)
\(\mathcal{D}_{FP}^2\)	\(\)	\(n\)