An Index for Ray Operators in 5d $E_n$ SCFTs

TL;DR

This paper constructs and analyzes an index for ray-supported BPS operators in 5d superconformal theories with exceptional symmetries, revealing full $E_n$ symmetry and novel charge properties of these operators.

Contribution

It introduces a new index for ray operators in 5d $E_n$ SCFTs, demonstrating full $E_n$ symmetry and exploring their charge and representation structure.

Findings

Index exhibits full $E_n$ symmetry at low order.

Ray operators carry nontrivial $ ext{Z}_{9-n}$ charge.

Leading index term is a minuscule $E_n$ representation.

Abstract

We construct an index for BPS operators supported on a ray in five dimensional superconformal field theories with exceptional global symmetries. We compute the $E_n$ representations (for $n=2,\dots,7$) of operators of low spin, thus verifying that while the expression for the index is only SO$(2n-2)\times$U(1) invariant, the index itself exhibits the full $E_n$ symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. For $n\le 7$, in contrast to local operators, they carry nontrivial charge under the $\mathbb{Z}_{9-n}\subset E_n$ center of the global symmetry. The representations that appear in the ray operator index are therefore different, for $n\le 7$, from those appearing in the previously computed superconformal index. For $3\le n\le 7$, we find that the leading term in the index is a character of a minuscule representation of $E_n$. We also discuss the case $n=8$, which presents a unique technical challenge, and remains an open problem.