====== Superconformal field theories ======

Superconformal field theories, or SCFTs, are theories whose spacetime symmetries include both the [[:conformal invariance|conformal group]] and some amount of [[:Supersymmetry|supersymmetry]].

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

===== List of SCFTs =====

{{topic>superconformal}}

===== Classification of superconformal algebras =====

A classification of possible superconformal algebras was given by [[https://inspirehep.net/literature/120988|Nahm]] in 1977. The outline of the argument is as follows:
  - 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
  - Prove that $L$ must be simple
  - The list of simple superalgebras is [[https://inspirehep.net/literature/125931|known]]. We select from the list the algebras satisfying 
    - $g_0$ contains the conformal algebra $so(d, 2)$ and 
    -  $g_1$ transforms in a spinorial representation of the conformal algebra $\mathfrak{so}(d,2)$, which is essentially the requirement that the supercharges are spinors.
  - 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 [[https://arxiv.org/pdf/hep-th/9712074.pdf|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.