Soft photon theorems from CFT Ward identites in the flat limit of AdS/CFT

TL;DR

This work provides a concrete framework to realize flat-space scattering within AdS/CFT by taking the large-radius limit and constructing CFT operators that create in/out scattering states. Through HKLL reconstruction and careful handling of boundary conditions, it derives explicit photon creation/annihilation operators from CFT data and shows that Weinberg soft photon theorems emerge as Ward identities of the dual CFT, including both electric and magnetic (SL(2,Z)) sectors. The analysis clarifies how asymptotic regions of Minkowski space map to fringe regions or complex-time domains in the CFT and highlights the role of Coulombic contributions in the full charge algebra. Together, these results bolster flat-space holography and illuminate how infrared symmetries of gauge theories are encoded in a lower-dimensional conformal framework, with promising avenues for extending to more general fields and gravity.

Abstract

S-matrix elements in flat space can be obtained from a large AdS-radius limit of certain CFT correlators. We present a method for constructing CFT operators which create incoming and outgoing scattering states in flat space. This is done by taking the flat limit of bulk operator reconstruction techniques. Using this method, we obtain explicit expressions for incoming and outgoing U(1) gauge fields. Weinberg soft photon theorems then follow from Ward identites of conserved CFT currents. In four bulk dimensions, gauge fields on AdS can be quantized with standard and alternative boundary conditions. Changing the quantization scheme corresponds to the S-transformation of SL(2,Z) electric-magnetic duality in the bulk. This allows us to derive both, the electric and magnetic soft photon theorems in flat space from CFT physics.

Figures (5)

• Figure 1: Penrose diagram of flat space. Boundary conditions for bulk field operators are imposed at the early (late) time Cauchy slices $\Sigma_{\pm}$. The region of space-time before $\Sigma_-$ or after $\Sigma_+$ is characterized by an asymptotic Hamiltonian $H_{\text{as}}$ that we will consider to be free.
• Figure 2: a) Penrose diagram of Anti-de Sitter space. Boundary conditions for normalizable bulk field operators can be imposed at the Cauchy slices $\Sigma_{\pm}^{\text{AdS}}$. The bulk path integral computes \ref{['eq:NormPhiZ']}. b) Gluing Lorentzian AdS to Euclidean half-spheres ${\cal M}_{\pm}$ and placing boundary conditions for the bulk field at ${\partial}{\cal M}_{\pm}$ yields the SvR prescription. This corresponds to a CFT expectation value involving coherent states.
• Figure 3: Section of the global AdS cylinder. The region inside the diamond corresponds to $\tau=t/L$ and $\rho=r/ L$ which turns flat as $L$ is taken to infinity. The Cauchy slices $\Sigma_{\pm}^{\prime}$ extend the Minkowski slices $\Sigma_{\pm}$ to global AdS space. Specifying a state on $\Sigma_{\pm}^{\prime}$ specifies a state in the conformal field theory, in the Cauchy slice $\Sigma_{\pm}^{\prime}\cap \partial\text{AdS}$. Alternatively, we can invoke asymptotic decoupling and use a free Hamiltonian in AdS to time evolve our fields to the Cauchy slices $\Sigma_{\pm}^{\text{AdS}}$. This would yield the computation of formula \ref{['eq:NormPhiZ']} with vanishing source ($\phi_0=0$), which is drawn in figure \ref{['fig:AdS']} a).
• Figure 4: The bulk field $\hat{\phi}(x)$ is placed inside a scattering region around the center of global AdS. The local operator can be reconstructed semi-classically in the boundary using HKLL. For the incoming field, the reconstruction involves the shaded blue part of the boundary spanning $\tau\in(-\pi,0)$. For the outgoing field, we choose the shaded red region spanning $\tau\in(0,\pi)$.
• Figure 5: The red fringe of global time can be parametrized by $\tau= {\pi \over 2}+{u\over L}$, where $u$ plays the role of retarded time at future null infinity ${\cal I}^+$. Similarly, the blue fringe parametrizes past null infinity ${\cal I}^-$. In the main text, we refer to these regions as $\tilde{\cal I}^{\pm}$. The shaded caps are analytic continuations of the boundary CFT in the imaginary direction of global time, and they play the role of future/past infinity $i^{\pm}$. In the main text we refer to these caps as $\partial {\cal M}_{\pm}$. The rest of the CFT can be interpreted as space-like infinity. As an example, an outgoing massless particle in flat space can be constructed in AdS/CFT by smearing an operator over a fringe of global time at ${\cal I}^+$.