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Real-time gauge/gravity duality

Kostas Skenderis, Balt C. van Rees

TL;DR

The paper addresses the need for a unitary, real-time holographic prescription to compute n-point functions in nontrivial states. It introduces a piecewise holographic framework that fills real-time contour segments with Lorentzian bulk solutions and imaginary segments with Euclidean solutions, joined by matching conditions at contour corners, encoding initial/final states in the Euclidean caps. The authors demonstrate the method by computing the vacuum-to-vacuum two-point function for a scalar in AdS3/CFT2, showing the expected iε prescription and time-ordering emerge from the contour and junction conditions. This real-time prescription extends holographic duality beyond Euclidean settings, enabling study of time-dependent phenomena and non-equilibrium states in strongly coupled systems.

Abstract

We present a general prescription for the holographic computation of real-time n-point functions in non-trivial states. In QFT such real-time computations involve a choice of a time contour in the complex time plane. The holographic prescription amounts to ``filling in'' this contour with bulk solutions: real segments of the contour are filled in with Lorentzian solutions while imaginary segments are filled in with Riemannian solutions and appropriate matching conditions are imposed at the corners of the contour. We illustrate the general discussion by computing the 2-point function of a scalar operator using this prescription and by showing that this leads to an unambiguous answer with the correct i epsilon insertions.

Real-time gauge/gravity duality

TL;DR

The paper addresses the need for a unitary, real-time holographic prescription to compute n-point functions in nontrivial states. It introduces a piecewise holographic framework that fills real-time contour segments with Lorentzian bulk solutions and imaginary segments with Euclidean solutions, joined by matching conditions at contour corners, encoding initial/final states in the Euclidean caps. The authors demonstrate the method by computing the vacuum-to-vacuum two-point function for a scalar in AdS3/CFT2, showing the expected iε prescription and time-ordering emerge from the contour and junction conditions. This real-time prescription extends holographic duality beyond Euclidean settings, enabling study of time-dependent phenomena and non-equilibrium states in strongly coupled systems.

Abstract

We present a general prescription for the holographic computation of real-time n-point functions in non-trivial states. In QFT such real-time computations involve a choice of a time contour in the complex time plane. The holographic prescription amounts to ``filling in'' this contour with bulk solutions: real segments of the contour are filled in with Lorentzian solutions while imaginary segments are filled in with Riemannian solutions and appropriate matching conditions are imposed at the corners of the contour. We illustrate the general discussion by computing the 2-point function of a scalar operator using this prescription and by showing that this leads to an unambiguous answer with the correct i epsilon insertions.

Paper Structure

This paper contains 4 sections, 16 equations, 3 figures.

Table of Contents

  1. Prescription.
  2. Example.
  3. Two-point function.
  4. Conclusion.

Figures (3)

  • Figure 1: The contour in the complex time plane used to obtain vacuum-vacuum amplitudes.
  • Figure 2: The $CFT_2$ contour, with the spatial circle and points at infinity added. Our prescription is to fill it in with an AdS spacetime consisting of three parts as well. The corners $C_\pm$ extend to hypersurfaces $S_\pm$ in the bulk, and $\delta M_L$ is the cylindrical conformal boundary of $M_L$.
  • Figure 3: Contours around the poles in the complex $\omega$-plane.