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

TL;DR

This work classifies 4D unitary conformal field theories with a unique stress tensor that also possess a higher-spin conserved current, proving that all correlators of symmetric currents must match those of a free field theory: either a free boson, a free fermion, or a free vector field. The authors develop the lightcone limit, spinor-helicity formalism, and quasi-bilocal operators to convert complex Ward identities into polynomial constraints with free-field solutions, establishing the existence of infinitely many higher-spin currents and fully constraining the theory to a free-field structure. The main result extends Maldacena and Zhiboedov’s 3D analysis to four dimensions and has implications for AdS/CFT duals involving Vasiliev-like higher-spin theories, while also clarifying the role of symmetry and current representations. Limitations include the focus on symmetric currents and the assumption of a single stress tensor; handling asymmetric currents or higher dimensions remains a nontrivial challenge requiring further techniques.

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].