Studying superconformal symmetry enhancement through indices

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