The gravity dual of real-time CFT at finite temperature

Abstract

We present a spherically symmetric aAdS gravity solution with SchwingerKeldyshboundary condition dual to a CFT at finite temperature defined on a complextime 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 piecesare interpreted as the gravity duals of the two Euclidean β/2 segments in the SK path, eachcoinciding with a Hartle-Hawking-Maldacena (TFD) vacuum state, while the Lorentzianregions naturally describes the real-time evolution of the TFD doubled system.Within the context of Skenderis and van Rees real-time holographic prescription, thenew solution should be compared to the Thermal AdS spacetime since both contributeto 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 bothbackgrounds and confront the results. When solving for the field we find that the gluingleads to a geometric realization of the Unruh trick via a completely holographic prescription.Interesting observations follow from hOLORi, which capture details of the entanglement ofthe (ground) state and the connectivity of the spacetime.