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Title: Integral geometry and holography

Abstract

We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS 3/CFT 2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS 3 whose kinematic space is two-dimensional de Sitter space.

Authors:
 [1];  [1];  [1];  [2]
+ Show Author Affiliations
  1. Stanford Univ., Stanford, CA (United States)
  2. SLAC National Accelerator Lab., Menlo Park, CA (United States)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1183697
Report Number(s):
SLAC-PUB-16293
Journal ID: ISSN 1029-8479; arXiv:1505.05515
Grant/Contract Number:  
AC02-76SF00515
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2015; Journal Issue: 10; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Theory-HEP; HEPTH; gauge-gravity correspondence; AdS-CFT correspondence

Citation Formats

Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, and Sully, James. Integral geometry and holography. United States: N. p., 2015. Web. doi:10.1007/JHEP10(2015)175.
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Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, & Sully, James. Integral geometry and holography. United States. https://doi.org/10.1007/JHEP10(2015)175
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Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, and Sully, James. Tue . "Integral geometry and holography". United States. https://doi.org/10.1007/JHEP10(2015)175. https://www.osti.gov/servlets/purl/1183697.
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@article{osti_1183697,
title = {Integral geometry and holography},
author = {Czech, Bartlomiej and Lamprou, Lampros and McCandlish, Samuel and Sully, James},
abstractNote = {We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.},
doi = {10.1007/JHEP10(2015)175},
url = {https://www.osti.gov/biblio/1183697}, journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 10,
volume = 2015,
place = {United States},
year = {2015},
month = {10}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1007/JHEP10(2015)175
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    Tensor networks from kinematic space
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    A stereoscopic look into the bulk
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    Entanglement, holography and causal diamonds
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    Solving the Schwarzian via the conformal bootstrap
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    The connection between holographic entanglement and complexity of purification
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    Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT
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    Entwinement in discretely gauged theories
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    On CFT and quantum chaos
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    Holographic spacetimes as quantum circuits of path-integrations
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    Complexity, action, and black holes
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    Remarks on the Sachdev-Ye-Kitaev model
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    Boundary dual of bulk local operators
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    Holographic de Sitter Geometry from Entanglement in Conformal Field Theory
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    Anti–de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
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    Einstein Equations from Varying Complexity
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    Conformal Quasicrystals and Holography
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    Entanglement wedge reconstruction and entanglement of purification
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