CFT Zoo

A classification of possible superconformal algebras was given by Nahm in 1977. The outline of the argument is as follows:

1. Consider a superalgebra $L = g_0 \oplus g_1$, where $g_0$ and $g_1$ denote the even and odd part of the superalgebra, respectively
2. Prove that $L$ must be simple
3. The list of simple superalgebras is known. We select from the list the algebras satisfying (a) $g_0$ contains the conformal algebra $so(d, 2)$ and (b) $g_1$ transforms in a spinorial representation of the SC
4. the result is the following classification: \begin{align} d = 3: & \qquad \mathfrak{osp}(\mathcal{N}|4) \ \supset \ \mathfrak{so}(3,2) \times \mathfrak{so}(\mathcal{N})_R \, , \\ d = 4: & \qquad \mathfrak{su}(2,2| \mathcal{N}) \ \supset \ \mathfrak{so}(4,2) \times \mathfrak{su}(\mathcal{N})_R \times \mathfrak{u}(1)_R \, , \\ d = 5: & \qquad \mathfrak{f}(4) \ \supset \ \mathfrak{so}(5,2) \times \mathfrak{su}(2)_R \, , (\text{requires} \mathcal{N} = 1) \\ d = 6: & \qquad \mathfrak{osp}(6,2| \mathcal{N}) \ \supset \ \end{align}

The algebras to the right of the $\supset$ denote the maximal bosonic subalgebra. We can see that they each contain the conformal algebra $\mathfrak{so}(d, 2)$, as required by (a). In addition to assumptions (a) and (b), this result relies on a sort of unitarity assumption– specifically, we must assume that there is a positive definite inner product on the algebra.