Causal structures and nonlocality in double holography

Published for SISSA by Springer Received: April 18, 2022 Accepted: June 17, 2022 Published: July 20, 2022 Causal structures and nonlocality in double holography a Department of Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan b Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan E-mail: , Abstract: Double holography plays a crucial role in recent studies of Hawking radiation and information paradox by relating an intermediate picture, in which a dynamical gravity living on an end-of-the-world brane is coupled to a non-gravitational heat bath, to a much better-understood BCFT picture as well as a bulk picture. In this paper, causal structures in generic double holographic setups are studied. We find that the causal structure in the bulk picture is compatible with causality in the BCFT picture, thanks to a generalization of the Gao-Wald theorem. On the other hand, consistency with the bulk causal structure requires the effective theory in the intermediate picture to contain a special type of superluminal and nonlocal effect which is significant at long range or IR. These are confirmed by both geometrical analysis and commutators of microscopic fields. Subregion correspondences in double holography are discussed with the knowledge of this nonlocality. Possible fundamental origins of this nonlocality and its difference with other types of nonlocality will also be discussed. Keywords: AdS-CFT Correspondence, Boundary Quantum Field Theory, Brane Dynamics in Gauge Theories ArXiv ePrint: 2107.01219 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2022)128 JHEP07(2022)128 Hidetoshi Omiyaa and Zixia Weib Contents 1 Introduction 1 2 Preliminaries and summary of technical results 2.1 Review of double holography and related topics 2.2 Compatibility of causality in the AdS/CFT correspondence 2.3 Summary of results: causal structures in double holography 4 4 6 9 10 10 14 4 Compatibility of causality in AdS/BCFT 4.1 An assumption on Σ 4.2 Showing statement B for generic configurations 4.3 Comments on more general Σ 17 17 18 20 5 Causal structure in the intermediate picture 5.1 Showing statement C for general configurations 5.2 Causal structure and relations to other setups 21 22 23 6 Commutators in the intermediate picture 24 7 Nonlocality and subregions in double holography 7.1 Breakdown of domain of dependence 7.2 Subregions and states associated to A 7.3 A tentative subregion duality and its breakdown 7.4 IR-sensitive nonlocality and quantum gravity 29 29 31 34 37 8 Conclusions and discussions 41 A AdSd+1 spacetime A.1 Coordinates A.2 Geodesics 42 42 43 1 Introduction The information loss problem in black hole evaporation [1–3] has puzzled physicists for decades and recently gets a great development [4, 5]. Starting from a pure initial state and tracking the time evolution from the formation to the evaporation of a black hole, the –1– JHEP07(2022)128 3 Vacuum configuration as an example 3.1 Vacuum configuration and null geodesics 3.2 Zooming in to the Poincaré patch –2– JHEP07(2022)128 entanglement entropy between its interior and exterior is expected, from unitarity, to start increasing from zero and finally return back to zero again. This behavior is known as a Page curve [2]. Hawking’s original computation [1] was performed with a local quantum field theory living on a classical spacetime with a black hole. By simply factorizing the Hilbert space into interior and exterior on the classical spacetime, Hawking’s computation suggests that the entanglement entropy monotonically grows and leads to a breakdown of unitarity, i.e. loss of information. Recent studies have resolved this problem by coupling a gravitational region containing a black hole to a non-gravitational region working as a heat bath, and studying the entanglement entropy between them. It is found that another saddle point which Hawking did not count dominates at late time and reproduces the expected Page curve [4, 5]. The existence and dominance of this saddle point is justified both by a class of doubly holographic models [6] and by gravitational path integral [7, 8]. Consider a d-dimensional AdS gravity living on Q interacting with a d-dimensional CFT living on Σ through a (d − 1)-dimensional interface ∂Q = ∂Σ. Double holography relates the current setup to two different but equivalent theories. One is a boundary CFT (BCFT) on Σ, which can be obtained by applying the AdS/CFT correspondence to Q and regarding it as a (d − 1)-dimensional CFT living on ∂Σ. The other one is an AdSd+1 gravity with an end-of-the-world brane floating in it. The asymptotic boundary and the end-of-the-world brane are identified with Σ and Q, respectively. In this paper, we call the latter two equivalent pictures the BCFT picture and the bulk picture respectively. At the same time, we call the original setup the intermediate picture, in the sense that it can be regarded as an intermediate process when jumping between the BCFT picture and the bulk picture. See figure 1 for a sketch. Although the terminology “double holography” is relatively new, the correspondence between the three pictures has been known for a long time since [9, 10]. In particular, the duality between the intermediate picture and the bulk picture is often called the KarchRandall type brane-world holography. On the other hand, the duality between the bulk picture and the BCFT picture is further explored in [11, 12] and called the AdS/BCFT correspondence. With double holography, the second saddle point in Hawking radiation can be thought to come from boundary OPE in the BCFT picture and minimal surfaces ending on the end-of-the-world brane in the bulk picture [13–15]. On the other hand, from a gravitational path integral point of view, this saddle point comes from spacetime configurations with higher topology [7, 8]. While dynamics in double holography plays a crucial role in recent studies of Hawking radiation, discussions in the Lorentzian signature [16–18] are limited and mostly focusing on specific spacetime configurations. One of the most important ingredients in a Lorentzian theory is the causal structure. In the following of this paper, we study the causal structures in double holography for generic spacetime dimensions and configurations. The causal structure in holography was firstly discussed by Gao and Wald [19]. They studied the AdS/CFT correspondence [20–22] and proved a theorem which implies that the causal structure in AdS is compatible with causality (the property that one cannot send a signal outside of the light cone) in CFT. As for double holography, we will firstly show bulk picture BCFT picture <latexit sha1_base64="4B+zTceAiF9S62RGDXb7AE9xkuo=">AAACb3ichVHLSsNAFD2N7/qqulAQpFgU3ZQbN4or0Y1LX61CK5LEUYemSUgmBS3+gB+gCxc+QET8DDf+gAs/QVxJBTcuvE0DoqLeY (...truncated)