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