Non-Perturbative Explorations of Chiral Rings in 4d $\mathcal{N}=2$ SCFTs

TL;DR

This paper investigates the presence and significance of often-overlooked $ar{ extbf{B}}$ multiplets in 4d $ extbf{N}=2$ SCFTs, revealing their ubiquity and connections to various phenomena through algebraic and topological methods.

Contribution

It demonstrates the widespread existence of $ar{ extbf{B}}$ multiplets in 4d $ extbf{N}=2$ SCFTs, including those with higher rank and specific anomalies, using algebraic and topological arguments.

Findings

$ar{ extbf{B}}$ multiplets are present in all local unitary $ extbf{N}>2$ SCFTs.

Existence of $ar{ extbf{B}}$ multiplets in theories with rank > 1 and conformal manifolds.

Broad class of theories with $ar{ extbf{B}}$ multiplets due to $ extbf{Z}_2$-valued 't Hooft anomalies.

Abstract

We study the conditions under which 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) have multiplets housing operators that are chiral with respect to an $\mathcal{N}=1$ subalgebra. Our main focus is on the set of often-ignored and relatively poorly understood $\overline{\mathcal{B}}$ representations. These multiplets typically evade direct detection by the most popular non-perturbative 4d $\mathcal{N}=2$ tools and correspondences. In spite of this fact, we demonstrate the ubiquity of $\overline{\mathcal{B}}$ multiplets and show they are associated with interesting phenomena. For example, we give a purely algebraic proof that they are present in all local unitary $\mathcal{N}>2$ SCFTs. We also show that $\overline{\mathcal{B}}$ multiplets exist in $\mathcal{N}=2$ theories with rank greater than one and a conformal manifold or a freely generated Coulomb branch. Using recent topological quantum field theory results, we argue that certain $\overline{\mathcal{B}}$ multiplets exist in broad classes of theories with the $\mathbb{Z}_2$-valued 't Hooft anomaly for $Sp(N)$ global symmetry. Motivated by these statements, we then study the question of whether $\overline{\mathcal{B}}$ multiplets exist in rank-one SCFTs with exactly $\mathcal{N}=2$ SUSY. We conclude with various open questions.