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tag:superconformal [2021/04/06 22:50]
Brian McPeak 		tag:superconformal [2021/04/10 06:23] (current)
Scott Lawrence [Superconformal field theories]
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 ====== Superconformal field theories ====== ====== Superconformal field theories ======
  
-Superconformal field theories, or SCFTs, are theories whose spacetime symmetries include both the [[Conformal Group|conformal group]] and some amount of [[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]].
  
 It follows from the classification given below that there are no SCFTs in dimension $d > 6$. It follows from the classification given below that there are no SCFTs in dimension $d > 6$.
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   - the result is the following classification: \begin{align}    - 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 = 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 = 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      d = 6: & \qquad \mathfrak{osp}(6,2| \mathcal{N}) \ \supset \      \mathfrak{so}(6,2) \times  \mathfrak{su}(2 \mathcal{N})_R     
  \end{align}  \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. 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.
Last modified: 2021/04/06 22:50
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