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

TL;DR

This work extends the five-dimensional $E_n$ SCFT program by constructing and computing an index for BPS ray operators, showing that the ray-index naturally organizes into $E_n$ representations despite being derived from an $SO(2n-2)\times U(1)$ embedding. The authors develop a framework linking ray operators to a Wilson-line partition function on $S^4\times S^1$, and they carefully incorporate zero-size instanton sectors via a D0-brane quantum mechanics with line defects, including a regulator through a D4$'$ brane. A key result is that ray operators carry nontrivial center charge ${\mathbb Z}_{9-n}$ for $3\le n\le 7$, and the leading ray-coefficient corresponds to minuscule $E_n$ representations, with explicit examples for $E_2$ through $E_7$; the $E_8$ case remains open due to technical hurdles in the instanton sector. Overall, the ray-index reveals representations not seen in the local operator SCI, offering a new window into the enhanced $E_n$ flavor symmetry and its role in the strong-coupling dynamics of 5d SCFTs, with potential links to 6d surface operators and geometric invariants.

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.

Figures (3)

• Figure 1: The location of the poles on the complex $\phi_1$ plane for instanton number $k=2$. The filled circles indicate the poles that are retained by the Jeffrey-Kirwan prescription, while the hollow circles indicate the poles that are ignored.
• Figure 2: The location of the poles on the $\phi_1-\phi_2$ real plane for instanton number $k=4$, for $\eta=(1,3)$. The lines are the loci where the argument of a single sinh in the denominator of the integrand vanishes. Poles are at the intersection of two lines. The solid circles indicate the poles that are retained by the Jeffrey-Kirwan prescription. (One pole, at $\phi_1=-M-\eM-2\eP$ and $\phi_2=M+\eP$, is outside the frame of the picture.) The hollow circles are possible locations of non-simple poles, where three lines intersect. (Whether they are simple or non-simple depends on $\text{Im}\phi_1$ and $\text{Im}\phi_2$.)
• Figure 3: The Dynkin diagram of $E_8$ and its subdiagram corresponding to SO(14) $\subset E_8$.