A Note on (Non)-Locality in Holographic Higher Spin Theories

TL;DR

This paper probes locality in holographic higher spin theories using Mellin amplitudes and reveals that the standard Mellin treatment of the free boundary four-point function is ill-defined. By defining the Mellin amplitude via linearity over constituent bulk processes and employing a deformed contour prescription, the authors show that the free theory Mellin amplitude can be taken as zero in a consistent sense, while the coordinate-space correlator remains nonzero, and that the singular part of the bulk scalar quartic vertex carries a distinctive pole structure. They argue that single-trace conformal blocks signal non-locality in the holographic higher spin setup, yet the peculiar Mellin-space cancellations suggest a nuanced notion of locality that may require deforming the Noether procedure. The results point toward a refined framework where bulk non-localities coexist with boundary consistency, potentially guiding deformations of locality notions and perturbative constructions of higher-spin theories in AdS or flat space.

Abstract

It was argued recently that the holographic higher spin theory features non-local interactions. We further elaborate on these results using the Mellin representation. The main difficulty previously encountered on this way is that the Mellin amplitude for the free theory correlator is ill-defined. To resolve this problem, instead of literally applying the standard definition, we propose to define this amplitude by linearity using decompositions, where each term has the associated Mellin amplitude well-defined. Up to a sign, the resulting amplitude is equal to the Mellin amplitude for the singular part of the quartic vertex in the bulk theory and, hence, can be used to analyze bulk locality. From this analysis we find that the scalar quartic self-interaction vertex in the holographic higher spin theory has a singularity of a special form, which can be distinguished from generic bulk exchanges. We briefly discuss the physical interpretation of such singularities and their relation to the Noether procedure.

Figures (2)

• Figure 1: This figure represents the real $(s,t,u)$ plane and locations of singularities of the reduced Mellin amplitude for the contact interaction. Singularities generated by $\Gamma(\delta_{ij})$ are shown as light-grey lines. The bold triangle is the real projection of the analyticity domain associated with the inverse Mellin transform of two variables, which defines admissible locations of the integration contour in the complex $(s,t,u)$ space. Depending on the way how we choose to close this integration contour at infinity, we pick one of the three sets of poles, enclosed in the red, the blue and the green contours.
• Figure 2: Here we illustrate the singularity structures of the reduced Mellin amplitudes for the contact Witten diagram and for exchanges with the field of dimension $\Delta$ in $s$, $t$ and $u$ channels. These are shown on figures $a)$, $b)$, $c)$ and $d)$ respectively. As before, bold triangles denote domains of analyticity of the reduced Mellin amplitude. Solid and empty circles represent locations of singularities associated with the three terms in the free theory correlator. Solid circles mean that the reduced Mellin amplitude has a given singularity, while empty circles mean that the reduced Mellin amplitude is regular at this point. The light-grey lines denote the leading single trace singularities of exchanges. For example, for the $s$-channel exchange it appears at $s=\Delta$.