Studying superconformal symmetry enhancement through indices

TL;DR

The paper develops a unified, index-based framework to diagnose SUSY enhancement across dimensions $d=3$--$6$ by classifying contributions of short superconformal multiplets into equivalence classes and mapping ${ m N}=1$ data to higher-${ m N}$ data in $d=4$ and $d=3$. It provides a rigorous set of necessary and sufficient conditions for enhancement in $d=4$ (with detailed subsections for ${ m N}=2$, ${ m N}=3$, and ${ m N}=4$ targets) and extends the analysis to 3d and 6d, including an AMS program that analyzes RG flows triggered by nilpotent vevs. The work offers practical, algebraically tractable criteria—often in terms of specific $t$-power coefficients in corrected indices—to test for enhancement, and applies these tools to glasses of theories (notably those studied by Agarwal–Maruyoshi–Song) to determine when UV ${ m N}=1$ theories flow to IR theories with higher SUSY. The results illuminate the constraints imposed by multiplet recombination, flavor currents, and protected operators on index data and provide a concrete methodology for verifying true SUSY enhancement in diverse SCFTs, including Argyres–Douglas theories in 4d and corresponding constructions in 6d.

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.