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Integral Geometry and Holography

Bartlomiej Czech
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Lampros Lamprou
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
Samuel McCandlish
(
- Stanford U., ITP and
- Stanford U., Phys. Dept.
)
,
James Sully
(
- SLAC
)

May 20, 2015

41 pages

Published in:

JHEP 10 (2015) 175

Published: Oct 27, 2015

e-Print:

1505.05515 [hep-th]

DOI:

10.1007/JHEP10(2015)175

Report number:

SU-ITP-15-07

View in:

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Abstract: (Springer)

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.

Note:

23 pages + appendices, including 23 figures and an exercise sheet with solutions; a Mathematica visualization tool

Gauge-gravity correspondence
AdS-CFT Correspondence
field theory: conformal
space: de Sitter
entropy: entanglement
duality: holography
dimension: 2
kinematics
geometry
anti-de Sitter

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