Abstract
In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in 3 ≤ d ≤ 6 must obey for the theory to have enhanced supersymmetry. We do that by noting that the index distinguishes a superconformal multiplet contribution to the index only up to a certain equivalence class it lies in. We classify the equivalence classes in d = 4 and build a correspondence between \( \mathcal{N}=1 \) and \( \mathcal{N}>1 \) equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the d = 4 \( \mathcal{N}=1 \) index for the theory to have \( \mathcal{N}>1 \) SUSY. We also find a necessary and sufficient condition on a d = 4 \( \mathcal{N}>1 \) index to correspond to a theory with \( \mathcal{N}>2 \). We then use our results to study some of the d = 4 theories described by Agarwal, Maruyoshi and Song, and find that the theories in question have only \( \mathcal{N}=1 \) SUSY despite having rational central charges. In d = 3 we classify the equivalence classes, and build a correspondence between \( \mathcal{N}>2 \) and \( \mathcal{N}>2 \) equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an \( 1\le \mathcal{N}\le 3 \) superconformal index in d = 3 to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an \( \mathcal{N}=4 \) index to correspond to an \( \mathcal{N}=4 \) theory. Finally, in d = 6 we find a necessary and sufficient condition for an \( \mathcal{N}=1 \) index to correspond to an \( \mathcal{N}>2 \) theory.
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Evtikhiev, M. Studying superconformal symmetry enhancement through indices. J. High Energ. Phys. 2018, 120 (2018). https://doi.org/10.1007/JHEP04(2018)120
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DOI: https://doi.org/10.1007/JHEP04(2018)120