Real-time gauge/gravity duality: Prescription, Renormalization and Examples

TL;DR

The paper develops a general real-time holographic prescription that extends gauge/gravity duality to Lorentzian dynamics and non-equilibrium states. It fills complex-time contours with bulk manifolds consisting of Lorentzian and Euclidean segments, applying matching conditions at corners and a robust holographic renormalization to produce finite, causal correlators. The approach is illustrated in detail for scalar fields and gravity, proving the cancellation of corner divergences and the continuity of one-point functions, and is then demonstrated across diverse backgrounds including global AdS, Poincaré AdS, BTZ black holes, and rotating BTZ. These results enable consistent derivations of time-ordered, retarded, and Wightman functions, including thermal and higher-point correlators, and lay groundwork for non-equilibrium holography and horizon-related phenomena.

Abstract

We present a comprehensive analysis of the prescription we recently put forward for the computation of real-time correlation functions using gauge/gravity duality. The prescription is valid for any holographic supergravity background and it naturally maps initial and final data in the bulk to initial and final states or density matrices in the field theory. We show in detail how the technique of holographic renormalization can be applied in this setting and we provide numerous illustrative examples, including the computation of time-ordered, Wightman and retarded 2-point functions in Poincare and global coordinates, thermal correlators and higher-point functions.

Figures (13)

• Figure 1: (a) Vacuum-to-vacuum contour. (b) In-in contour. (c) Real-time thermal contour.
• Figure 2: A single corner in the contour in the complex time plane. We use this part of a field theory contour to illustrate the holographic renormalization.
• Figure 3: On the left, the dotted lines represent two bulk hypersurfaces given by $t = f$ in the vicinity of a corner in the boundary contour at $t=0$. On the right, we see that around a full turn in the boundary contour it is natural to expect that $f_{[1]} = -f_{[0]}$.
• Figure 4: On the left, various contours in the complex time plane. The vertical segments in the first two contours should be thought of as extending to infinity, yielding a vacuum state on the corner. The circles in the third contour should be identified; it is then a thermal contour. The crosses represent an example of the operator insertions we consider. On the right, we sketch the spacetimes consisting of piecewise Euclidean and Lorentzian AdS$_3$ that fill the given contours. One should impose matching conditions on the hypersurfaces between the segments.
• Figure 5: The dots represent poles in complex frequency space and the curved line a Feynman contour that avoids them.