List of Series of Fusion Rings
The list below contains series of interesting fusion rings, mainly associated with conformal field theorys and/or affine Lie algebras. Clicking on a name will redirect you to the page of the respective series of fusion rings. Here the categorifiability is considered over the complex field. For the last 5 columns a value of True indicates that there exists at least 1 way to categorify the fusion ring to a category with the respective structure, for all values of the parameter of the series. There can be multiple categories stemming from the same fusion ring with multiple (possibly disjunct) properties. The abreviations FC, PFC, UFC, BFC, MFC respectively stand for Fusion Category, Pivotal Fusion Category, Unitary Fusion Category, Braided Fusion Category, Modular Fusion Category (most likely, all series will be MTCs, so we could/should drop the True statements alltogether).
| Name | Condition | Rank | \(\mathcal{D}_{FP}^2\) | FC | PFC | UFC | BFC | MFC |
|---|---|---|---|---|---|---|---|---|
| \(\mathbb{Z}_n\) | \(n\geq 1\) | \(n\) | \(n\) | ✓ | ✓ | ✓ | ✓ | ✓ |
| \(su(2)_k\) | \(k\geq 1\) | \(k+1\) | \(\frac{k+2}{2\sin^2(\frac{\pi}{k+2})}\) | ✓ | ✓ | ✓ | ✓ | ✓ |
| \(su(n)_2\) | \(n \geq 2\) | \(\frac{n(n+1)}{2}\) | \(\frac{n(n+2)}{4\sin^2(\frac{\pi}{n+2})}\) | ✓ | ✓ | ✓ | ✓ | ✓ |
| \(so(2p+1)_2\) | \(p\geq 1\) | \(p+4\) | \(8p+4\) | ✓ | ✓ | ✓ | ✓ | ✓ |
| \(so(2p)_2\) | \(p\geq 1\) | \(p+7\) | \(8p\) | ✓ | ✓ | ✓ | ✓ | ✓ |