The gravity dual of real-time CFT at finite temperature

Published for SISSA by Springer Received: August 31, 2018 Accepted: November 8, 2018 Published: November 21, 2018 Marcelo Botta-Cantcheff, Pedro J. Martı́nez and Guillermo A. Silva Instituto de Fı́sica de La Plata, CCT La Plata, CONICET & Departamento de Fı́sica, Universidad Nacional de La Plata, C.C. 67, La Plata, 1900 Argentina E-mail: , , Abstract: We present a spherically symmetric aAdS gravity solution with SchwingerKeldysh 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 2pt 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 hOL OR i, which capture details of the entanglement of the (ground) state and the connectivity of the spacetime. Keywords: AdS-CFT Correspondence, Black Holes, Thermal Field Theory ArXiv ePrint: 1808.10306 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2018)129 JHEP11(2018)129 The gravity dual of real-time CFT at finite temperature Contents 1 2 Holography for a closed time-contour and TFD 2.1 The TFD formulation and the in/out scenario 2.2 The Schwinger-Keldysh boundary problem in the large N limit 2 4 7 3 Building up the gravity solution 3.1 Continuity conditions 8 8 4 Bulk massive scalar field 4.1 Two-sided black hole geometry 4.2 Thermal AdS geometry 10 10 13 5 Bulk correlators 15 6 Conclusions 18 A Propagator in configuration space 19 1 Introduction The study of the gravity/gauge correspondence at finite temperature was initiated by Witten in [1], where the CFT was formulated in periodic imaginary time, i.e. the Matsubara formalism. In this setup, the conformal field theory (CFT) is formulated on the S d−1 × Sβ1 boundary. Since the work of Hawking-Page [2], two aAdS gravity solutions are known to fulfill the boundary conditions: the so-called Thermal AdS and the Euclidean AdS black hole, which dominate in the low temperature and high temperature limit respectively. However, a real time extension of the formalism is needed for the study of nonequilibrium and finite temperature dynamical processes. The initial steps in this direction started with [3] (see also [4, 5]) where, with a Schwinger-Keldysh perspective, the CFT finite temperature propagator matrix elements were reproduced using the maximally extended AdS-BH geometry. Nevertheless, the procedure involved imposing infalling boundary conditions at the horizon, which contrasts with the holographic viewpoint. In [6] (see [7–10] for previous work), a Lorentzian formulation of the correspondence was presented. By gluing Euclidean and Lorentzian regions, the prescription relied only on boundary data without resorting to boundary conditions inside the bulk. Within this setup, the Thermal AdS real time extension was easily built [11]. The high temperature CFT matrix elements were re-obtained, but at the expense of requiring two copies of the maximally extended AdS black hole [11, 12]. –1– JHEP11(2018)129 1 Introduction 2 Holography for a closed time-contour and TFD This section is devoted to elaborate on the SvR prescription when considering a conformal field theory defined on a closed time contour in the complex plane which involves two imaginary-time intervals (of length β/2) as shown in figure 1a [13–15]. It will be argued how the external sources on these intervals univocally define states of the CFT, which can be described as pure bra/kets in the TFD framework so that the CFT path integral describes a in/out scattering process. Finally, we will discuss the gravitational dual to the thermal SK path in the large N approximation, and present a new solution involving a two-sided black holes that solves the boundary problem and dominates over Thermal AdS –2– JHEP11(2018)129 The Skenderis-van Rees (SvR) prescription [6, 11], provides the natural framework to study gravity duals to Schwinger-Keldysh (SK) closed paths [13–16]. Among the possible closed path in the complex t-plane, Umezawa singled out a particular one which connects the SK and Thermofield Dynamics (TFD) formalisms [15, 17, 18]. It is worth mentioning that already in [3] the authors stress that their prescription acquires nice properties for the SK contour highlighted by Umezawa. It is this particular path that we will elaborate on in this work. The TFD interpretation is central in our study and it has already proved useful in the AdS/CFT context in [19] where, based in the Hartle Hawking construction [20], Maldacena showed that the half Euclidean black hole geometry maps to the TFD vacuum state in the boundary field theory. See [21, 22] for other works on maximally extended AdS black holes. In this work, we present an exact spherically symmetric solution to Einstein gravity dual to the CFT on a SK path mentioned above. It is genuinely holographic geometry in the sense that is completely determined by asymptotic boundary data, and in addition, is the natural real-time extension of the Euclidean AdS-BH by inserting the two-sided exterior of a single black hole. This is in line with Israel’s interpretation of the TFD degrees of freedom as being physically realized within the two-sides of BH geometry [23]. The present solution can also be thought of as the real time evolution of the Hartle-Hawking-Maldacena state [19] under the TFD Hamiltonian. This work is organized as follows: in section 2, we review the SvR prescription and its relation to SK thermal paths, i.e. closed time contours. We will describe the CFT theory defined by our contour in the TFD picture and study the predictions for the bulk theory via the duality. We will consider the large N limit and the two contributions to the saddle point approximation of the gravitational path integral, noting that the path admits also a second Thermal-AdS dual which we should compare our results with. In section 3 we describe in detail the construction of the geometry as well as the boundary conditions for the field inhabiting our geometry. Section 4 will deal with the KG field computations inside our geometry and the gluing procedure. For completeness in section (...truncated)