Spinning Witten Diagrams

TL;DR

This work presents a unified framework to decompose tree-level spinning four-point Witten diagrams into conformal partial waves, using a combination of shadow formalism and harmonic analysis in AdS. By diagonalising the map between boundary spinning three-point structures and bulk cubic couplings, the authors enable a holographic reconstruction of all cubic interactions for totally symmetric fields, and they apply the method to the type A higher-spin theory dual to the free $O(N)$ model. The approach cleanly separates single-trace and double-trace contributions in CPWEs and provides explicit prescriptions for calculating CPWEs of generic spinning exchanges, including off-shell cubic couplings and higher-spin propagators. The results advance holographic bootstrap perspectives and offer practical tools for analyzing locality and bulk interactions in higher-spin holography.

Abstract

We develop a systematic framework to compute the conformal partial wave expansions (CPWEs) of tree-level four-point Witten diagrams with totally symmetric external fields of arbitrary mass and integer spin in AdS$_{d+1}$. Employing this framework, we determine the CPWE of a generic exchange Witten diagram with spinning exchanged field. As an intermediate step, we diagonalise the linear map between spinning three-point conformal structures and spinning cubic couplings in AdS. As a concrete application, we compute all exchange diagrams in the type A higher-spin gauge theory on AdS$_{d+1}$, which is conjectured to be dual to the free scalar $O\left(N\right)$ model. Given a CFT$_d$, our results provide the complete holographic reconstruction of all cubic couplings involving totally symmetric fields in the putative dual theory on AdS$_{d+1}$.