Studying superconformal symmetry enhancement through indices

TL;DR

This paper classifies the conditions on superconformal indices across dimensions 3 to 6 that indicate supersymmetry enhancement, providing a systematic way to identify theories with higher SUSY from their indices.

Contribution

It introduces a classification of index equivalence classes and establishes necessary and sufficient conditions for supersymmetry enhancement in various dimensions.

Findings

Classified index equivalence classes in 4, 3, and 6 dimensions.

Derived conditions for supersymmetry enhancement from indices.

Applied results to specific 4D theories, confirming their SUSY levels.

Abstract

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.