Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange

TL;DR

This work develops a comprehensive, gauge-aware framework to analyze higher-spin exchanges in AdS$_{d+1}$ and their holographic imprint on boundary scalar four-point functions. By deriving complete massless spin-$s$ bulk-to-bulk propagators in the metric-like formalism across de Donder, traceless, and manifest-trace gauges and employing ambient space methods plus the split representation, the authors recast spin-$s$ exchanges into a conformal partial wave expansion on the boundary. The four-point exchange between scalar pairs is computed in detail and expressed as sums over boundary CPWE contributions with explicitly determined coefficients, enabling direct comparison with CFT data and the eventual reconstruction of the bulk quartic vertex. The study also illuminates how cubic-improvement terms, though on-shell trivial, feed into quartic contact structures, underscoring subtle aspects of locality in higher-spin holography and outlining pathways for future work to establish bulk locality more firmly.

Abstract

Within holography, we calculate the contribution of an arbitrary spin-s gauge boson exchange in $AdS_{d+1}$ to the four-point function with scalar operators on the boundary. As an important ingredient, we first compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges. The split representation of the bulk-to-bulk propagators in terms of bulk-to-boundary propagators allows to present the higher-spin exchange diagram in the form of a conformal partial wave expansion. Our results provide a step towards the larger goal of the holographic reconstruction of bulk interactions, and of clarifying bulk locality.

Figures (6)

• Figure 1: Total four-point Witten diagrams contributing to the bulk computation of the four-point function of the scalar singlet bilinear operator $\mathcal{O}_{\Delta}$ in the $O\left(N\right)$ vector model. Here, $\Delta$ is the dimension of the bulk scalar dual to $\mathcal{O}_{\Delta}$, and the $P_i$ are fixed points on the boundary of AdS$_{d+1}$. In the present paper, we compute diagram 1.(a) (the "s-channel") for a spin-$s$ gauge boson $\varphi_s$. Computations for the remaining exchange channels 1.(b) and 1.(c) follow in the same way. Diagram 1.(d) illustrates the quartic scalar contact interaction.
• Figure 2: Use of the split representation for the bulk-to-bulk propagators in the exchange results in a decomposition in terms of products of three-point Witten diagrams, involving a pair of real scalar fields and a field whose spin, $s-\ell$, is summed over. The common boundary point of the three-point functions is integrated over. This is to be compared with the definition \ref{['cpwe']} of the conformal partial wave expansion in conformal field theory.
• Figure 3: Visualisation of the ambient space with space-like axes $X^a$$\left(a=1,...d\right)$, and light-like axes $X^+$ and $X^-$. The hypercone $X^2=0$ is in green and the hyperboloid $X^2=-L^2$ is in blue. The paraboloid is the Poincaré section obtained by intersecting the light cone with a light-like hyperplane at constant $X^+$, and is illustrated in the figure for $X^+=1$.
• Figure 4: Split representation of AdS harmonic function $\Omega_{\nu,\ell}$ as a product of two spin-$\ell$ bulk-to-boundary propagators with dimensions $\tfrac{d}{2}\pm i\nu$. The boundary point $P$ is integrated over.
• Figure 5: Four-point exchange in the "s-channel" of a spin-$s$ gauge boson between two pairs of real scalars $\phi$, which have dual CFT operator $\mathcal{O}_{\Delta}$.