Superconformal field theories

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

It follows from the classification given below that there are no SCFTs in dimension $d > 6$.

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
   1. $g_0$ contains the conformal algebra $so(d, 2)$ and
   2. $g_1$ transforms in a spinorial representation of the conformal algebra $\mathfrak{so}(d,2)$, which is essentially the requirement that the supercharges are spinors.
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 \, , \ \, \qquad \qquad \mathcal{N} \neq 4 \\ & \qquad \mathfrak{psu}(2,2| 4) \ \supset \ \mathfrak{so}(4,2) \times \mathfrak{su}(4)_R \, , \ \, \qquad \qquad \qquad \qquad \mathcal{N} = 4\\ % d = 5: & \qquad \mathfrak{f}(4) \ \supset \ \mathfrak{so}(5,2) \times \mathfrak{su}(2)_R \qquad \qquad \qquad \qquad \qquad \qquad \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.