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--- superconformal_field_theory [2021/04/06 21:49]
Brian McPeak
+++ — (current)
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-==== 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 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
-  - the result is the following classification: \begin{align}
-d = 3: & \qquad \mathfrak{osp}(\mathcal{N}|4) \\
-d = 4: & \qquad \\
-d = 5: & \qquad \\
-d = 6: & \qquad
- \end{align}
-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.

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