Lorentzian AdS, Wormholes and Holography

TL;DR

The paper investigates Lorentzian AdS wormholes in five-dimensional Einstein-Gauss-Bonnet gravity and their holographic duals on two asymptotic AdS boundaries. It extends the GKPW prescription to multi-boundary geometries by treating independent boundary sources on each boundary and constructing two bulk-boundary propagators, enabling the computation of both same-boundary and cross-boundary two-point functions. Through AdS2 and explicit wormhole examples, it shows that cross-boundary correlators can arise from either entanglement or genuine coupling between the boundary theories, and proposes a geometric criterion—based on Euclidean topology—to separate these contributions. The results illuminate how spacetime connectivity and causal structure are encoded in the dual QFT action, with implications for emergent spacetime and quantum gravity research.

Abstract

We investigate the structure of two point functions for the QFT dual to an asymptotically Lorentzian AdS-wormhole. The bulk geometry is a solution of 5-dimensional second order Einstein Gauss Bonnet gravity and causally connects two asymptotically AdS space times. We revisit the GKPW prescription for computing two-point correlation functions for dual QFT operators O in Lorentzian signature and we propose to express the bulk fields in terms of the independent boundary values phi_0^\pm at each of the two asymptotic AdS regions, along the way we exhibit how the ambiguity of normalizable modes in the bulk, related to initial and final states, show up in the computations. The independent boundary values are interpreted as sources for dual operators O^\pm and we argue that, apart from the possibility of entanglement, there exists a coupling between the degrees of freedom leaving at each boundary. The AdS_(1+1) geometry is also discussed in view of its similar boundary structure. Based on the analysis, we propose a very simple geometric criterium to distinguish coupling from entanglement effects among the two set of degrees of freedom associated to each of the disconnected parts of the boundary.

Figures (2)

• Figure 1: Contours in $\bf \omega$ complex plane: when performing the $\omega$ integration in \ref{['deltatt']} any arbitrary contour (depicted as red) can be deformed to be the Feynman contour (depicted in blue) plus contributions from encircling the poles \ref{['polos']}. The encircling of positive (negative) frequency poles fix the initial (final) states $\psi_{\mathrm {i,f}}$ in \ref{['2ptf']}.
• Figure 2: (a) Poincare patch of Lorentzian AdS: the value of the scalar field $\phi$ at any point in the bulk depends not only on the boundary value $\phi_0$ but also on the value of the field at the future and past horizons $\phi_H$. (b) Lorentzian wormhole geometry: the value of the scalar field $\phi$ at any point in the bulk depends not only on two boundary values $\phi^\pm_0$ (blue) but also on normalizable modes in the bulk $\phi^{(\sf n)}$ (red). The dependence, in both pictures, on the normalizable modes (red lines) correlates to a choice of the initial and final states $|\psi_{\mathrm {i,f}}\rangle$ and manifest as a choice of contour in the complex $\omega$ plane when computing the correlator \ref{['deltatt']}.