Flat space physics from AdS/CFT

TL;DR

The paper develops a concrete bridge from CFT correlators to flat-space scattering amplitudes by taking a flat AdS radius limit of HKLL-reconstructed bulk operators and Fourier transforming to momentum space. It demonstrates that S-matrix elements for massless and massive external particles can be obtained from CFT2 correlators and reproduce the expected BMS3-invariant structures in 2+1 dimensions. By analyzing scattering on conical deficits and deficit-state correlators, the authors show that the flat-limit correspondence extends to nontrivial backgrounds, validating the proposed map in physically interesting settings. This provides an indirect microscopic holographic framework for flat-space gravity and suggests avenues for generalization to higher dimensions and deeper connections to BMS symmetry and bulk locality.

Abstract

We propose a formula relating scattering S-matrix amplitudes to correlators of a conformal field theory. The proposal implements a flat limit of the field theory, providing an indirect microscopic description of gravitational theories with asymptotically flat boundary conditions. The formula is valid for both massive and massless external particles, and reduces to existing expressions in the literature when all particles are either simultaneously massless or massive. We test the result in various (2+1)-dimensional examples such as simple BMS3 invariant correlators and blocks. We also study two-point correlators in conformal field theory deficit states to obtain known expressions for non-trivial scattering in asymptotically flat conical geometries.

Figures (3)

• Figure 1: Construction of flat space scattering states from states in a conformal field theory. a) The bulk field $\hat{\phi}(x)$ is placed inside a scattering region around the center of global AdS (blue). The local operator can be reconstructed semi-classically in the boundary using the HKLL formula \ref{['eq:HKLL']}. The reconstruction involves the shaded red part of the boundary, which is space-like separated to the scattering region. b) After performing a Fourier transform with respect to the flat space coordinates and implementing a flat limit, the reconstruction is performed by placing a primary operator in a Euclidean continuation of the boundary manifold. If the flat space particle is massless, the CFT operator is inserted at a real value of global time $\tau=\pm \pi/2$.
• Figure 2: Pictorial representation of the S-matrix according to formula \ref{['eq:Smatrix']}. We have chosen to represent a $2\rightarrow 2$ event where two "in" particles of any mass are created in the scattering region and become two "out" particles after interacting non-trivially in flat space. The interaction is represented by a red blob, and its details are encoded in the relevant CFT correlator in the form of a kinematic divergence. The primary operators are located at insertions of the form \ref{['eq:insertion']}, whose complex value has been represented as an insertion in the Euclidean half-sphere.
• Figure 3: Free particles/fields ( red) scattering around a cone geometry ( blue). The cone is represented in blue, with a dashed blue branch-cut representing the lack of periodicity under $\phi\sim\phi+2\pi$. a) Deflection of the trajectory of a classical particle that propagates freely in the conical deficit geometry. b) Quantum wave-packets approach the cone source at $t=-\infty$, initially unaffected by the presence of the conical deficit. c) As the waves interact with the source of the cone, they scatter. Parts remains a plane-wave ( red) that ignore the non-trivial geometry, while some of the wave scatters spherically ( green) away from the location of the source.