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--- tag:superconformal [2021/04/05 05:55]
Scott Lawrence [Superconformal field theories]
+++ tag:superconformal [2021/04/10 06:23] (current)
Scott Lawrence [Superconformal field theories]
@@ Line 1: / Line 1: @@
 ====== Superconformal field theories ======
-Superconformal field theories are those CFTs that possess [[Supersymmetry|supersymmetry]].
+Superconformal field theories, or SCFTs, are theories whose spacetime symmetries include both the [[:conformal invariance|conformal group]] and some amount of [[:Supersymmetry|supersymmetry]].
-Minwalla [[https://arxiv.org/pdf/hep-th/9712074.pdf|showed]] that there are no SCFTs in dimension $d > 6$.
+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.

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