From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories

TL;DR

This work clarifies how a broad class of bulk Lorentzian geometries can be reconstructed, at linear order, from states defined by Euclidean path integrals with local operator sources in holographic CFTs. By combining the Skenderis–van Rees Schwinger–Keldysh construction with a perturbative bulk analysis, it shows a concrete map between Euclidean sources and Lorentzian initial data for both scalar and metric perturbations, and verifies the correspondence with direct CFT one-point functions. A central finding is that, while arbitrary smooth initial data can be generated perturbatively, extreme localization requires progressively smaller amplitudes to keep the expansion valid, and nonperturbative arguments reveal that Lorentzian data cannot produce spacelike Fourier modes in one-point functions. The results illuminate the limits of representing classical bulk configurations via Euclidean-path-integral states and point to interesting extensions to mixed states, subregion physics, and beyond-linear-order effects. These insights contribute to understanding the emergence of bulk geometry from CFT data and the holographic encoding of initial conditions for gravity.

Abstract

We consider states of holographic conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the extent to which arbitrary bulk coherent states can be represented by such Euclidean path-integrals in the CFT. We construct the associated dual Lorentzian spacetimes perturbatively in the sources. Extending earlier work, we provide explicit formulae for the Lorentzian fields to first order in the sources for general scalar field and metric perturbations in arbitrary dimensions. We check the results by holographically computing the Lorentzian one-point functions for the sourced operators and comparing with a direct CFT calculation. We present evidence that at the linearized level, arbitrary bulk initial data profiles can be generated by an appropriate choice of Euclidean sources. However, in order to produce initial data that is very localized, the amplitude must be taken small at the same time otherwise the required sources diverge, invalidating the perturbative approach.

Figures (3)

• Figure 1: We prepare a state by turning on sources in Euclidean time $\tau$. This gives us some initial data on the $\tau=t=0$ surface in the bulk, which we can obtain from the bulk to boundary propagator (blue) and it's time derivative. Then, we can further evolve this data in real-time $t$ using the Lorentzian propagator (red) to obtain the real-time asymptotics, from which we read off the CFT one-point functions. Spatial directions in the CFT are supressed.
• Figure 2: Path integral contours for evaluation of one point functions in Lorentzian time (first figure) and Euclidean time (second figure). Red contours indicate where sources have been turned on. To compute one-point functions at Euclidean times in a neighborhood of $\tau=0$ where the sources vanish, we can use a single Euclidean contour with $\tau \in (-\infty,\infty)$.
• Figure 3: Plot of $z_0^2 \left(V_{min}\right)^{-1}$ vs $N$ on a semi-log scale, where $V_{min}$ is the minimum variance of an initial data function and $N$ is the ratio of the $L_2$ norm of sources to the $L_2$ norm of the initial data function. This figure uses $\omega_{max} z_0=40$ and $\epsilon z_0=\frac{1}{5}$ and plots the results for values of $\omega_{min} z_0$ listed in the legend. The curve suggests that the minimum variance approaches zero as the norm of the sources is increased for any choice of $\omega_{min} z_0$.