The gravity dual of real-time CFT at finite temperature

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 β/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 〈O L O R 〉, which capture details of the entanglement of the (ground) state and the connectivity of the spacetime.

Holographic excited states in AdS black holes

“…The symmetric Schwinger-Keldysh path involving two imaginary-time intervals (of equal length β/2) [18,32], shown in Fig. 2(a), was investigated in [17] in the holographic context.…”

“…The evolution operator in this case factorizes as U = U R U F U L U I , where U L/R are ordinary real time evolution operators with the CFT Hamiltonian H deformed by external (local) sources φ(x, t ). The operators U I ,F on both imaginary time intervals, univocally describe the initial/final excited states in terms of the local sources φ(x, τ) [17]:…”

“…where the Euclidean time τ runs on the intervals (−β/2, 0) and (0, β/2) on I and F respectively. In fact, in the TFD context these operators are equivalent to pure states, rearranged as kets in the duplicated states space [17]. Let us consider a conformal field theory, whose states belong to the Hilbert space H .…”

“…In a recent work [17], we presented a novel geometry dual to a Schwinger-Keldysh (SK) contour [18] describing real time evolution of a finite temperature CFT in which standard Thermo Field Dynamics (TFD) [19] computations can be carried holographically. We studied the geometry, its two-point functions and its role in the context of the Hawking-Page (HP) transition.…”

“…a semi-infinite Euclidean path integral with non-zero sources corresponded to a precise holographic state, coherent in the large-N limit. In this work we pursue an analogous objective for the geometry we built in [17]. Its TFD interpretation will provide the required In-Out structure.…”