Superconformal field theories, or scfts, are theories whose spacetime symmetries include both the conformal group and some amount of supersymmetry.

superconformal

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 conformal algebra $\mathfrak{so}(d,2)$
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 \quad ( \text{requires } \mathcal{N} = 1) \,
d = 6: & \qquad \mathfrak{osp}(6,2| \mathcal{N}) \ \supset \ \mathfrak{so}(6,2) \times \mathfrak{su}(2 \mathcal{N})_R \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). The fact that the list truncates at $d = 6$ is essentially due to assumption $(b)$. Of all the simple superalgebras, only $\mathfrak{f}(4)$ has an odd part which transforms in a spinorial representation. However, in low dimension, there exist exceptional isomorphisms between lie algebras. This allows, for example, the vector of $\mathfrak{usp}(4)$ to be interpreted as the spinor of $\mathfrak{so}(5)$, which is exactly what is required to ensure that the odd part of the $d = 3$ algebra is spinorial (see section 4 of Minwalla for further discussion on this point. 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.