Holographic excited states in AdS black holes

Published for SISSA by Springer Received: January 17, 2019 Accepted: March 29, 2019 Published: April 3, 2019 Marcelo Botta-Cantcheff, Pedro J. Martı́nez and Guillermo A. Silva Instituto de Fı́sica de La Plata, CCT La Plata — CONICET, and Departamento de Fı́sica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina E-mail: , , Abstract: We have recently presented a geometry dual to a Schwinger-Keldysh closed time contour, with two equal β/2 length Euclidean sections, which can be thought of as dual to the Thermo Field Dynamics formulation of the boundary CFT. In this work we study non-perturbative holographic excitations of the thermal vacuum by turning on asymptotic Euclidean sources. In the large-N approximation the states are found to be thermal coherent states and we manage to compute its eigenvalues. We pay special attention to the high temperature regime where the manifold is built from pieces of Euclidean and Lorentzian black hole geometries. In this case, the real time segments of the Schwinger-Keldysh contour get connected by an Einstein-Rosen wormhole through the bulk, which we identify as the exterior of a single maximally extended black hole. The Thermal-AdS case is also considered but, the Lorentzian regions become disconnected, its results mostly follows from the zero temperature case. Keywords: AdS-CFT Correspondence, Black Holes, Thermal Field Theory ArXiv ePrint: 1901.00505 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)028 JHEP04(2019)028 Holographic excited states in AdS black holes Contents 1 2 The SvR approach, excited states, and In-Out formalism 2.1 Brief review of the In-Out formalism: open paths 2.2 Closed paths: the Schwinger-Keldysh contour and TFD 2.2.1 TFD evolution and transition amplitudes 2.2.2 Piecewise holographic map 3 3 6 8 9 3 Excited states from the bulk perspective 3.1 Bulk geometry and gluing conditions 3.2 Scalar field solution 3.2.1 Lorentzian sources 3.2.2 Euclidean sources 3.2.3 Analyticity through the wormhole 3.3 Results from bulk analysis 11 11 12 13 15 16 17 4 Canonical quantization of the bulk fields and BDHM dictionary 4.1 Canonical quantization of scalar fields in a BH geometry 4.2 BDHM at finite temperature, TFD doubling and coherence 4.3 On the Unruh’s trick in the TFD formulation 19 19 20 22 5 Discussion and conclusions 23 A Low temperature excited states: thermal AdS 25 1 Introduction AdS/CFT [1] is mostly developed in Euclidean time [2, 3]. Conceptually, there is no fundamental principle forcing an Euclidean formulation of the duality. However, a direct approach to real time holography give raise to subtleties [4]. In particular, real time evolution demands initial and final conditions which are not immediate to characterize from both sides of the duality, in conflict with a strict holographic viewpoint. The Skenderis and van Rees (SvR) prescription [5, 6] provides a completely holographic real time extension of the GKPW standard prescription [2, 3]. It essentially maps the initial/final state information, through auxiliary Euclidean regions, to boundary data in the CFT. The general set-up of the SvR prescription thus deals with manifolds of mixed signature, the philosophy being to require only holographic/boundary data. In the SvR framework, sources on the Lorentzian asymptotic boundary are thought of as devices to obtain n-point correlation functions. On the other hand, Euclidean sources –1– JHEP04(2019)028 1 Introduction –2– JHEP04(2019)028 play a very different role and prepare the state of the system at a given time. The foundational works [5, 6], for example, showed that turned off Euclidean sources prepare the vacuum state. Turned on sources, in turn, allow to prepare (holographic) excited states of the CFT [7], see also [8] for related work. In [7] we began the study of general non-trivial sources in manifolds with mixed signature complementing the bulk treatment with the BDHM dictionary [9]. With this machinery we were able to show that, in the large N approximation, the excited states obtained by turning on Euclidean sources are coherent states. Interacting fields in the bulk lead to states of a modified nature, which we analyzed in [10]. A more systematic understanding of these excitations is under development. These holographic excited states have been of interest in recent literature [11]–[16]. In a recent work [17], we presented a novel geometry dual to a Schwinger-Keldysh (SK) contour [18, 19] describing real time evolution of a finite temperature CFT in which standard Thermo Field Dynamics (TFD) [20] computations can be carried holographically. We studied the geometry, its two-point functions and its role in the context of the HawkingPage (HP) transition. For comparison, in this same work we studied the real time extension of Thermal-AdS. The main objective of the present work is to study holographic excited states on these finite temperature geometries. We will provide a review of the formalism developed in [7] and derive its extension to the finite temperature set-up. The most relevant result of [7] that we will exploit in the present paper is the In-Out formulation, that allowed to split and interpret the Euclidean and Lorentzian path integral pieces as initial/final states and real time evolution of the system respectively. This splitting permitted us to study of the excited states as objects (kets) independently of the precise SK path it is glued to, e.g. 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. Previous thermal geometries [6, 22, 23] were not suitable for this interpretation. We will compute inner products and matrix elements of CFT local operators for holographic excited states, the latter directly related to linear response quantities in standard TFD formalism. The inner products, which require collapsing the real time segments, can be understood as a reinterpretation of standard Euclidean result with non-zero sources. The kernels in these objects, due to the coherent nature of the excited states, define Kähler potential in the space of states which may be of interest for the developments in [24]. The matrix elements on their own help to recognize the thermal coherent character of the states and to determine its eigenvalues. The path integral approach demands finding a general solution to the equations of motion with non-trivial Euclidean and Lorentzian sources turned on. We will build it in detail, checking that CFT information is enough to give a unique analytic solution inside the bulk. This result is non-trivial once we notice that the Lorentzian Rindler-like patches, dual to real time evolutions, end up being glued analytically through an Einstein-Rosen (E (...truncated)