Higher-spin Interactions from CFT: The Complete Cubic Couplings

TL;DR

The paper tackles the problem of obtaining a complete cubic-couplings catalog for the Type A minimal bosonic higher-spin theory in $AdS_{d+1}$ by holographically reconstructing bulk interactions from boundary data. It develops a generating-function framework for conserved currents, derives explicit two- and three-point OPE coefficients in the dual free $O(N)$ vector model, and computes tree-level three-point Witten diagrams for generic cubic vertices using ambient-space methods. The main result is a closed-form bulk action $\mathcal{V}=\sum g_{s_1,s_2,s_3} I^{0,0,0}_{s_1,s_2,s_3}$ with $g_{s_1,s_2,s_3}$ given explicitly, reproducing the boundary correlators and matching flat-space limits in $AdS_4$. Together, these findings provide a gauge-consistent holographic dictionary between CFT current correlators and higher-spin bulk couplings, advancing the program of HS holography.

Abstract

In this letter we provide a complete holographic reconstruction of the cubic couplings in the minimal bosonic higher-spin theory in AdS$_{d+1}$. For this purpose we also determine the OPE coefficients of all single-trace conserved currents in the $d$-dimensional free scalar $O\left(N\right)$ vector model, and compute the tree-level three-point Witten diagram amplitudes for a generic cubic interaction of higher-spin gauge fields in the metric-like formulation.

Figures (1)

• Figure 1: Euclidean AdS and its boundary in ambient space. This figure displays the AdS surface $X^2 = - R^2=-1$ and the identification of a (green) boundary point with a (green) light ray of the light cone $P^2 = 0$, which intersects the Poincaré section \ref{['ps']} on a (red) point