On Locality, Holography and Unfolding

TL;DR

The paper proposes a concrete locality criterion for higher-spin theories in AdS by decomposing pseudo-local cubic couplings into primary canonical currents and improvements, and requiring the induced series of overlap coefficients to converge. It develops ambient-space and unfolding formalisms to compute the relevant coefficients $C^{(s)}_l$ and analyzes 3d and 4d examples, showing that Vasiliev’s Schwinger-Fock gauge generally yields nonlocal tails that fail the criterion. When the criterion is satisfied, pseudo-local tails can be resummed into a finite primary-current piece that reproduces AdS/CFT correlators, as demonstrated by explicit 3d Witten-diagram computations aligned with CFT bilinear forms. The work clarifies how bulk locality can be defined in higher-spin theories, identifies gauge-frame issues in known formulations, and lays groundwork for controlled quartic-order locality analyses and HS-symmetric regularisations.

Abstract

We study the functional class and locality problems in the context of higher-spin theories and Vasiliev's equations. A locality criterion that is sufficient to make higher-spin theories well-defined as field theories on Anti-de-Sitter space is proposed. This criterion identifies admissible pseudo-local field redefinitions which preserve AdS/CFT correlation functions as we check in the 3d example. Implications of this analysis for known higher-spin theories are discussed. We also check that the cubic coupling coefficients previously fixed in 3d at the action level give the correct CFT correlation functions upon computing the corresponding Witten diagrams.