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 N=1 $$ \mathcal{N}=1 $$ and N>1 $$ \mathcal{N}>1 $$ equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the d = 4 N=1 $$ \mathcal{N}=1 $$ index for the theory to have N>1 $$ \mathcal{N}>1 $$ SUSY. We also find a necessary and sufficient condition on a d = 4 N>1 $$ \mathcal{N}>1 $$ index to correspond to a theory with N>2 $$ \mathcal{N}>2 $$. 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 N=1 $$ \mathcal{N}=1 $$ SUSY despite having rational central charges. In d = 3 we classify the equivalence classes, and build a correspondence between N>2 $$ \mathcal{N}>2 $$ and N>2 $$ \mathcal{N}>2 $$ equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an 1≤N≤3 $$ 1\le \mathcal{N}\le 3 $$ superconformal index in d = 3 to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an N=4 $$ \mathcal{N}=4 $$ index to correspond to an N=4 $$ \mathcal{N}=4 $$ theory. Finally, in d = 6 we find a necessary and sufficient condition for an N=1 $$ \mathcal{N}=1 $$ index to correspond to an N>2 $$ \mathcal{N}>2 $$ theory