Vanishing short multiplets in rank one 4d/5d SCFTs

Abstract

Abstract We study the short multiplet spectrum in 4d $$ \mathcal{N} $$ N = 2 superconformal theories of low rank using the full superconformal indices and the selection rules from the superconformal representation theory. We find a universal expression for the leading terms for the superconformal index of rank one H0, H1, H2, D4, E6, E7 theories. From this result, we argue that certain short multiplets appear in the operator product expansions involving stress-tensor, conserved current, and Coulomb branch operator vanish. We also apply the same procedure to 5d superconformal theories and find that E1 theory has vanishing short multiplets analogous to that of the H1 theory.