Local bulk S-matrix elements and CFT singularities

TL;DR

This work investigates how bulk local physics, encoded in bulk $S$-matrix elements, can be retrieved from boundary CFT data in the AdS/CFT framework by focusing on the plane-wave limit and a distinctive Lorentzian singularity in four-point functions. It constructs boundary sources that produce localized bulk wavepackets and shows that, in a controlled double-scaling limit, the bulk flat-space $S$-matrix emerges from boundary correlators via a precise correspondence between the singularity structure at $z\!=\!\bar{z}$ and the reduced transition amplitude $iT(s,t)$. The authors verify the mechanism with concrete examples from bulk supergravity—including contact interactions, scalar exchange, and graviton exchange—recovering known flat-space matrix elements and providing explicit normalization factors and functional dependencies. Overall, the paper offers a diagnostic route to test whether a given CFT can reproduce bulk locality, and outlines how the approach could be extended to loop, string, and higher-spin regimes while identifying the key role of the $z=\bar{z}$ singularity in encoding bulk energy-momentum conservation.

Abstract

We give a procedure for deriving certain bulk S-matrix elements from corresponding boundary correlators. These are computed in the plane wave limit, via an explicit construction of certain boundary sources that give bulk wavepackets. A critical role is played by a specific singular behavior of the lorentzian boundary correlators. It is shown in examples how correlators derived from the bulk supergravity exhibit the appropriate singular structure, and reproduce the corresponding S-matrix elements. This construction thus provides a nontrivial test for whether a given boundary conformal field theory can reproduce bulk physics, and where it does, supplies a prescription to extract bulk S-matrix elements in the plane wave limit.

Figures (3)

• Figure 1: AdS$_2$ is shown in blue. The revolution axis of the hyperboloid corresponds to the spacelike direction of $\mathbb{R}^{2,1}$ and the transverse plane is timelike. Global time is the angular coordinate in this plane. The point $X_0$ is a reference point in AdS$_2$. The null momenta $k_1$ and $k_2$ live in the tangent space to AdS at $X_0$. The boundary sources are supported in the neighborhood of the boundary points $P_i$. On the right, we show the universal cover of AdS$_2$ conformally compactified.
• Figure 2: Sketch of the boundary points configuration in AdS$_{d+1}$ for the Lorentzian kinematical condition of equal cross ratios $z=\bar{z}$. Here, all such points are lightlike related to the bulk point $x_0$.
• Figure 3: Complex paths $z(\alpha)$ and $\bar{z}(\alpha)$ starting from the Euclidean regime at $\alpha=0$ to the Lorentzian one at $\alpha=\frac{\pi}{2}$, for the particular scattering angle $\Theta=1$.