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

TL;DR

The paper proves that any unitary conformal field theory in dimensions greater than three with a unique stress tensor and at least one higher-spin conserved current must have an infinite tower of higher-spin currents and whose correlators coincide with those of a free theory: either n free bosons, n free fermions, or n free (d-2)/2-forms. By introducing lightcone limits and constructing quasi-bilocal operators that reproduce free-field OPE behavior on the lightcone, the authors transform complex Ward identities into constraints that uniquely determine all correlators to match one of the three free theories. They demonstrate that the three-point stress-tensor function is fixed to one of the free structures, and, under unitarity, all higher-point functions follow suit, effectively classifying higher-spin CFTs in d>3. The results have implications for AdS/CFT dualities with Vasiliev gravity and imply quantization of the effective number of free fields, while noting caveats in odd dimensions and nonunitary cases.

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]