On time-dependent AdS/CFT

TL;DR

The paper tackles the challenge of formulating a Lorentzian, time-dependent AdS/CFT dictionary by deriving and connecting bulk propagators (Feynman, retarded, advanced) to boundary correlators through a careful treatment of horizon boundary conditions and radial regularization. It develops a dual description in terms of a 1D radial Green function, linking the bulk dynamics to boundary data via a precise relation $g(u,u') = u^{\frac{d-1}{2}} {u'}^{\frac{d-1}{2}} k_{\lambda}(u,u')$ and explicit radial propagators, then constructs bulk-to-boundary propagators with a finite radial cutoff $u_0$ that respect the correct causal structure. The work computes near-boundary and subleading behaviors, establishing the mapping between leading sources $\phi_-$ and subleading vevs $\phi_+$ and derives both Feynman and retarded boundary two-point functions, showing how the closed-time-path formalism naturally appears in the holographic setting. Overall, it provides foundational steps toward a complete real-time AdS/CFT dictionary and highlights the UV/IR interplay between boundary causality and radial dynamics, with potential applications to time-dependent gravity processes and strongly coupled field theories at large $N$.

Abstract

We clarify aspects of the holographic AdS/CFT correspondence that are typical of Lorentzian signature, to lay the foundation for a treatment of time-dependent gravity and conformal field theory phenomena. We provide a derivation of bulk-to-boundary propagators associated to advanced, retarded and Feynman bulk propagators, and provide a better understanding of the boundary conditions satisfied by the bulk fields at the horizon. We interpret the subleading behavior of the wavefunctions in terms of specific vacuum expectation values, and compute two-point functions in our framework. We connect our bulk methods to the closed time path formalism in the boundary field theory.