Integral geometry and holography

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James

doi:10.1007/JHEP10(2015)175

Title: Integral geometry and holography

Journal Article · 27 October 2015 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/JHEP10(2015)175 · OSTI ID:1183697

Czech, Bartlomiej ^[1]; Lamprou, Lampros ^[1]; McCandlish, Samuel ^[1]; Sully, James ^[2]

Stanford Univ., Stanford, CA (United States)
SLAC National Accelerator Lab., Menlo Park, CA (United States)

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.

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Research Organization:: SLAC National Accelerator Lab., Menlo Park, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

Grant/Contract Number:: AC02-76SF00515

OSTI ID:: 1183697

Report Number(s):: SLAC-PUB-16293; arXiv:1505.05515

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 10 Vol. 2015; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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Holographic Spacetimes as Quantum Circuits of Path-Integrations Takayanagi, Tadashi arXiv https://doi.org/10.48550/arxiv.1808.09072	text	January 2018
Holographic relations for OPE blocks in excited states Cresswell, Jesse C.; Jardine, Ian T.; Peet, Amanda W. arXiv https://doi.org/10.48550/arxiv.1809.09107	text	January 2018
Holographic description of boundary gravitons in (3+1) dimensions Asante, Seth K.; Dittrich, Bianca; Haggard, Hal M. arXiv https://doi.org/10.48550/arxiv.1811.11744	text	January 2018
Holographic entanglement contour, bit threads, and the entanglement tsunami Kudler-Flam, Jonah; MacCormack, Ian; Ryu, Shinsei arXiv https://doi.org/10.48550/arxiv.1902.04654	text	January 2019

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