Constraining conformal field theories with a higher spin symmetry

TL;DR

This paper demonstrates that the presence of a single higher spin conserved current in a 3D conformal field theory enforces the theory to be equivalent to a free field theory, extending the Coleman-Mandula theorem to CFTs.

Contribution

It proves that a single higher spin conserved current implies an infinite tower of such currents, constraining the theory to be free, thus extending symmetry constraints to conformal field theories.

Findings

Presence of one higher spin current implies infinite conserved currents.

Correlation functions match those of free bosons or fermions.

Higher spin symmetry constrains CFTs to be free theories.

Abstract

We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. This is an extension of the Coleman-Mandula theorem to CFT's, which do not have a conventional S matrix. We also briefly discuss the case where the higher spin symmetries are "slightly" broken.