Rindler bulk reconstruction and subregion duality in AdS/CFT

Published for SISSA by Springer Received: August 8, 2022 Accepted: October 24, 2022 Published: November 9, 2022 Sotaro Sugishitaa,b and Seiji Terashimac a Institute for Advanced Research, Nagoya University, Nagoya, Aichi 464-8601, Japan b Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan c Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: , Abstract: In this paper, we study the AdS-Rindler reconstruction. The CFT operators naively given by the holographic dictionary for the AdS-Rindler reconstruction contain tachyonic modes, which are inconsistent with the causality and unitarity of the CFT. Therefore, the subregion duality and the entanglement wedge reconstruction do not hold. We also find that the tachyonic modes in the AdS-Rindler patch lead to arbitrary highenergy or trans-Planckian modes in the global AdS. It means that the mode expansion of the Rindler patch is sensitive to the UV limit of the theory, that is, quantum gravity. In addition, the tachyonic modes are related to the existence of null geodesics connecting the past and future horizons. Keywords: AdS-CFT Correspondence, 1/N Expansion, Models of Quantum Gravity ArXiv ePrint: 2207.06455 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2022)041 JHEP11(2022)041 Rindler bulk reconstruction and subregion duality in AdS/CFT Contents 1 2 Review of AdS-Rindler 2.1 Coordinates 2.2 Free scalar fields in AdS-Rindler patch 2 2 5 3 Incompleteness of AdS-Rindler bulk reconstruction 7 4 Null geodesics in the AdS-Rindler patch 16 5 View from the global AdS 19 1 Introduction and summary It is important to study how the bulk gravitational theory emerges from the CFT in the AdS/CFT correspondence in order to understand what is the spacetime in the quantum gravity. An explicit realization of this for the bulk fields is called the bulk reconstruction and has been studied, for example, in [1–5], in particular for the free bulk theory limit. A basic question of the bulk reconstruction is what is the reconstructable bulk fields from CFT operators supported only in a subregion of the boundary spacetime. The subregion duality [6–12] claims that the bulk fields supported in a subregion of the bulk spacetime, called the entanglement wedge, can be reconstructed from CFT operators on the boundary subregion, but the bulk fields outside it cannot be reconstructed. Here the boundary subregion for the CFT operators corresponds to a boundary limit of the bulk subregion. This reconstruction is called the entanglement wedge reconstruction and assumed to be correct in many studies although it is claimed to be incorrect in [13, 14]. In particular, for the Rindler patch of the AdS spacetime, the explicit bulk reconstruction formula was given in [3] for the free bulk theory limit. In this AdS-Rindler reconstruction, the boundary limit of the free scalar field on bulk AdS-Rindler is naively identified to the CFT primary operator by the BDHM formula [15]. In this paper, we study the AdS-Rindler reconstruction and find that the naive identification by the BDHM formula is inconsistent. Indeed, the CFT operators naively given by the BDHM formula for the AdS-Rindler reconstruction contain tachyonic modes, which are inconsistent with the causality and unitarity of the CFT although these modes are consistent as the bulk theory.1 Here, the important ingredient of this conclusion is that we consider the large, but finite N CFT. Thus, the Planck length (over the AdS-scale) is arbitrary small, but finite. This means that this inconsistency comes from the truly 1 It is also argued in [16] that a difficulty in the bulk reconstruction arising from tachyonic modes in black hole backgrounds (where the modes are called evanescent modes). In [17], it is also discussed that such tachyonic modes are related to the ill-definedness of the smearing functions in the bulk reconstruction. –1– JHEP11(2022)041 1 Introduction and summary 2 Review of AdS-Rindler In this section, we will review the Rindler patch in the AdS spacetime and the free scalar fields on it. Some references on the Rindler patch in the AdS/CFT are [25–28]. 2.1 Coordinates We summarize the coordinates of AdSd+1 used in this paper. Using the embedding coordinates into R2,d , AdSd+1 is described as  − X −1 2  − X0 2  + X1 2 –2–  + · · · + Xd 2 = −1. (2.1) JHEP11(2022)041 non-perturbative effects of the quantum gravity. The free bulk theory, which corresponds to the generalized free CFT, should be modified above the Planck energy because such a state becomes a black hole and the free spectrum around the fixed background is no longer valid. In the bulk point of view, there seem to be no problems to consider the mode expansion in the AdS-Rindler patch. However, we show that the tachyonic modes in the AdS-Rindler patch correspond to arbitrary high energy modes of the global AdS, for example, the transPlanckian modes. This means that the mode expansion of the Rindler patch is sensitive to the UV completion of the theory which is the quantum gravity in our case. In other words, the low energy modes of the AdS-Rindler patch do not correspond to the low energy modes of the global AdS. Therefore the subregion duality does not hold and the AdS-Rindler reconstruction is incomplete. It is an important question which part of bulk local fields cannot be reconstructed from the CFT operators in the Rindler patch. In the AdS-Rindler patch there are null geodesics never reaching the asymptotic boundary. This type of null geodesics starts from the past AdS-Rindler horizon and ends on the future one. We show that the non-reconstructable tachyonic modes are related to these horizon-horizon geodesics. Instead of using the AdS-Rindler coordinates, we can study which part of the bulk local operators are able to be reconstructed by CFT operators in a subregion from the global AdS (and the corresponding CFT on the cylinder) viewpoint. Indeed, in [13, 14] using the bulk reconstruction developed in [5, 18, 19], such studies had been done. The results obtained in this paper are perfectly consistent with the studies in [13, 14]. We believe that the results in this paper are substantial ingredients for the understanding of spacetime in the AdS/CFT and the quantum gravity. We emphasize that the low energy description of the bulk theory with the AdS-Rindler quantization should be modified in the AdS/CFT. This is interesting because it is often believed that the low energy description is valid even in the Rindler coordinate with the horizon because there is no curvature singularity. We expect that such a violation is an essential property of (black hole) horizon because it is due to the behavior of fields near the horizon, which is universal to general black hole horizons not restricted to the Rindler one. This violation might b (...truncated)