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

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 [1]. This paper supersedes the previous paper by the authors [2].

Universal bounds on operator dimensions from the average null energy condition

“…form (1.4) with n > 2 imply that the correlators of the energy-momentum tensor coincide with those of the free theory [5,6]. The classification of unitary representations in (1.2) reveals that there are possible conserved currents beyond the symmetric ones described above.…”

“…When k = 7, the bound may be saturated by the free field operator F F ∂F demonstrating that the result (1.10) is optimal. When k = 6, the bound cannot be saturated by free fields and it is unclear whether operators of spin (6,1) exist in the range 6 ≤ ∆ < 13/2. Given the form of our (k, 0) and (k, 1) results, it is tempting to conjecture a general formula for the gap in the spectrum of anomalous dimensions of general Lorentz representations implied by the average null energy condition.…”

“…Specifically, we must integrate along y − to produce the average null energy operator, and we must Fourier transform the external states to give them definite energy. 6 The integral along y − is straightforward to evaluate using the residue theorem once we restore the imaginary parts from (2.17):…”

“…(2.23) 6 Note that these states, like all states of definite momentum, are delta function normalized. Therefore although we have two operators h and h † , only one Fourier transform is required.…”

A Note on Rectangular Partially Massless Fields

“…Indeed, it was shown in [44] that if a 3-dimensional CFT which is unitary, obeys the cluster decomposition axiom and has a (unique) Lorentz covariant stress-tensor plus at least one higher-spin current, then this theory is either a CFT of free scalars or free spinors. This was generalised to arbitrary dimensions in [45,46] 3 , where the authors showed that this result holds in dimensions d 3, up to the additional possibility of a free CFT of ( d−2…”