Constraining conformal field theories with a higher spin symmetry in d> 3 dimensions

TL;DR

This paper proves that in higher-dimensional conformal field theories with higher-spin symmetries, correlators are highly constrained and must resemble free theories or specific forms, extending previous 3D results.

Contribution

It generalizes the classification of conformal field theories with higher-spin symmetry to dimensions greater than three, showing they are essentially free or of specific form.

Findings

Higher-spin conserved currents imply free or specific structured correlators.

In even dimensions, all three structures (free bosons, fermions, forms) can exist.

In odd dimensions, the form structure may not exist, and its realization is uncertain.

Abstract

We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in d>3 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 imply that the correlators of the stress tensor and the conserved currents of the theory must coincide with one of the following three possibilities: a) a theory of n free bosons (for some integer n), b) a theory of n free fermions, or c) a theory of n (d-2)/2-forms. For d even, all three structures exist, but for d odd, it may be the case that the third structure (c) does not; if it does exist, it is unclear what theory, if any, realizes it. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [arXiv:1112.1016]. This paper supersedes the previous paper by the authors [arXiv:1307.8092]