Spinning AdS Propagators

TL;DR

The paper develops an embedding-space framework to treat symmetric traceless tensors in AdS, enabling concise construction of bulk-to-bulk propagators for massive spin-J fields and a robust split representation in terms of boundary data. This representation facilitates the conformal partial wave decomposition of Witten diagrams and yields explicit Mellin amplitudes for graviton exchange between scalars in general dimensions. Key results include complete spin-1 and spin-2 split representations (up to contact terms), a graviton split representation in the massless limit consistent with Ward identities, and a precise mapping between bulk cubic couplings and CFT OPE coefficients. The methods streamline higher-spin AdS computations and provide exact, checkable expressions for CPW expansions and Mellin amplitudes with potential extensions to broader tensorial and spacetime settings.

Abstract

We develop the embedding formalism to describe symmetric traceless tensors in Anti-de Sitter space. We use this formalism to construct the bulk-to-bulk propagator of massive spin J fields and check that it has the expected short distance and massless limits. We also and a split representation for the bulk-to-bulk propagator, by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. We exemplify the use of this representation with the computation of the conformal partial wave decomposition of Witten diagrams. In particular, we determine the Mellin amplitude associated to AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude.

Figures (6)

• Figure 1: Euclidean AdS and its boundary in the embedding space. This picture shows the $AdS_2$ surface $X^2=-1$ and the identification of a boundary point (in blue) with a light ray (in red) of the light cone $P^2=0$, which intersects the Poincaré section on a (black) point.
• Figure 2: Representation of AdS harmonic function $\Omega_{\nu,J}$ in terms of two spin $J$ bulk-to-boundary propagators of dimension $h\pm i\nu$ integrated over the boundary point.
• Figure 3: The split representation of the spin $J$ propagator obtained by integrating over $\nu$ and summing over the spin $l$ of two bulk-to-boundary propagators of dimension $h\pm i\nu$ integrated over the boundary point, according to (\ref{['eq:SlipRepStart']}).
• Figure 4: Witten diagram that computes a CFT three-point function of a spin $J$ primary operator of dimension $\Delta$ and two scalar primary operators of dimension $\Delta_1$ and $\Delta_2$.
• Figure 5: Witten diagram describing a spin $J$ exchange between scalar primaries of arbitrary dimension. Using the split representation of the bulk-to-bulk propagator this diagram can be converted into the product of two three-point functions integrated over the common boundary point $P_5$.