The Gravity Dual of Real-Time CFT at Finite Temperature

TL;DR

This paper constructs a gravity dual for a finite-temperature CFT on a complex time contour, using a novel AdS black hole geometry that models real-time evolution and entanglement, and compares correlators in different backgrounds.

Contribution

It introduces a new spherically symmetric AdS black hole solution with Schwinger-Keldysh boundary conditions, linking Euclidean and Lorentzian regions to model real-time finite-temperature dynamics.

Findings

Computed scalar two-point functions in the new geometry.

Demonstrated the geometric realization of the Unruh trick holographically.

Analyzed entanglement and spacetime connectivity through correlator behavior.

Abstract

We present a spherically symmetric aAdS gravity solution with Schwinger-Keldysh boundary condition dual to a CFT at finite temperature defined on a complex time contour. The geometry is built by gluing the exterior of a two-sided AdS Black Hole, the (aAdS) Einstein-Rosen wormhole, with two Euclidean black hole halves. These pieces are interpreted as the gravity duals of the two Euclidean $\beta/2$ segments in the SK path, each coinciding with a Hartle-Hawking-Maldacena (TFD) vacuum state, while the Lorentzian regions naturally describes the real-time evolution of the TFD doubled system. Within the context of Skenderis and van Rees real-time holographic prescription, the new solution should be compared to the Thermal AdS spacetime since both contribute to the gravitational path integral. In this framework, we compute the time ordered 2-pt functions of scalar CFT operators via a non-back-reacting Klein-Gordon field for both backgrounds and confront the results. When solving for the field we find that the gluing leads to a geometric realization of the Unruh trick via a completely holographic prescription. Interesting observations follow from $\langle {\cal O}_L{\cal O}_R\rangle$, which capture details of the entanglement of the (ground) state and the connectivity of the spacetime.