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

TL;DR

This paper investigates the spectrum of short multiplets in rank one 4d and 5d superconformal field theories using superconformal indices, revealing universal patterns and the vanishing of certain multiplets in operator product expansions.

Contribution

It provides a universal expression for superconformal indices in rank one theories and shows the vanishing of specific short multiplets in OPEs, extending to 5d theories.

Findings

Universal superconformal index expressions for rank one theories.

Certain short multiplets vanish in OPEs involving key operators.

E1 5d theory exhibits vanishing short multiplets similar to H1.

Abstract

We study the short multiplet spectrum in 4d $\mathcal{N}=2$ 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 $H_0, H_1, H_2, D_4, E_6, E_7$ 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 $E_1$ theory has vanishing short multiplets analogous to that of the $H_1$ theory.