A note on ${\cal N}\ge 6$ Superconformal Quantum Field Theories in three dimensions

TL;DR

This note investigates global symmetry content and supersymmetry enhancement in $\mathcal{N}\ge 6$ three-dimensional SCFTs using only the superconformal algebra, unitarity, and the stress-tensor multiplet. It proves that irreducible $\mathcal{N}=6$ theories host exactly one conserved $U(1)$ current within the stress-tensor multiplet. It further shows that any $\mathcal{N}=7$ SCFT is actually $\mathcal{N}=8$, and that $\mathcal{N}=8$ theories have no global symmetries. These results explain empirical observations in ABJM/BLG constructions and clarify why purely $\mathcal{N}=7$ theories have not been found.

Abstract

Based on the structure of the three-dimensional superconformal algebra we show that every irreducible ${\mathcal N}=6$ three-dimensional superconformal theory containes exactly one conserved U(1)-symmetry current in the stress tensor supermultiplet and that superconformal symmetry of every ${\mathcal N}=7$ superconformal theory is in fact enhanced to ${\mathcal N}=8$. Moreover, an irreducible ${\cal N}=8$ superconformal theory does not have any global symmetries. The first observation explains why all known examples of ${\mathcal N}=6$ superconformal theories have a global abelian symmetry.