Studying superconformal symmetry enhancement through indices

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 1le 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.