A modular toolkit for bulk reconstruction
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Springer
Received: March 22, 2019 Accepted: April 11, 2019 Published: April 18, 2019
Thomas Faulkner,a Min Lia and Huajia Wanga,b a
Department of Physics, University of Illinois, 1110 W. Green St., Urbana IL 61801-3080, U.S.A. b Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We introduce new tools for studying modular flow in AdS/CFT. These tools allow us to efficiently extract bulk information related to causality and locality. For example, we discuss the relation between analyticity in modular time and entanglement wedge nesting which can then be used to extract the location of the Ryu-Takayanagi (RT) surface directly from the boundary theory. Probing the RT surface close to the boundary our results reduce to the recent proof of the Quantum Null Energy Condition. We focus on heavy probe operators whose correlation functions are determined by spacelike geodesics. These geodesics interplay with the RT surface via a set of rules that we conjecture and give evidence for using the replica trick. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1806.10560
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)119
JHEP04(2019)119
A modular toolkit for bulk reconstruction
Contents 1 Introduction
1 4 8 11
3 Applications 3.1 Mirrors for mirror operators 3.2 Entanglement wedge nesting 3.3 The Quantum Null Energy Condition
12 13 14 18
4 Justification of rules 4.1 ∆1/2 4.2 Modular flow replica trick
20 20 22
5 Discussion
28
A Re-deriving the QNEC results A.1 Example: geodesic in pure AdS A.2 General state
28 29 31
1
Introduction
A promising approach to quantum gravity comes from a class of dualities where gravity emerges from non-gravitational degrees of freedom that are inherently quantum mechanical yet in principle under control non-perturbatively [1, 2]. The quantum information aspects of these degrees of freedom provide a direct way to reveal the emergence of gravity, thus connecting the paradigm of emergence to BH thermodynamics [3, 4] and the entropic nature of Einstein’s equations [5]. Making these statements precise is an ongoing challenge. Most optimistically we might be able to use an improved understanding of this paradigm to liberate ourselves from the models that are most under control — that of quantum gravity in asymptotically AdS spaces. Toy models, which are for the most part lattice based quantum systems, that provide a playground for these ideas have been recently developed highlighting the entanglement structure [6] and quantum error correcting aspects of holography elucidated in [7]. Examples include tensor networks such as MERA [8] and holographic error correcting codes [9, 10]. However including dynamics in these toy models is less clear, and an especially difficult problem involves finding models with emergent locality on the fine-grained or sub-AdS scale.
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JHEP04(2019)119
2 Some rule
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