Constraining conformal field theories with a higher spin symmetry in d=4

TL;DR

This paper proves that four-dimensional unitary conformal field theories with higher-spin symmetries are essentially free theories, as their correlators are fixed to those of free bosons, fermions, or vectors by Ward identities.

Contribution

It generalizes the 3D higher-spin symmetry constraints to 4D, showing such theories are completely determined and equivalent to free field theories.

Findings

Higher-spin currents are infinite in 4D theories with such symmetries.

Ward identities fix correlators to free theory forms.

The result extends Maldacena and Zhiboedov's 3D findings to 4D.

Abstract

We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in four dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents suffice to completely fix the correlators of the stress tensor and the conserved currents to be equal to one of three free field theories: the free boson, the free fermion, and the free vector field. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [arXiv:1112.1016].