Integral Geometry and Holography
Abstract
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS/CFT 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 whose kinematic space is two-dimensional de Sitter space.
Cite
@article{arxiv.1505.05515,
title = {Integral Geometry and Holography},
author = {Bartlomiej Czech and Lampros Lamprou and Samuel McCandlish and James Sully},
journal= {arXiv preprint arXiv:1505.05515},
year = {2015}
}
Comments
23 pages + appendices, including 23 figures and an exercise sheet with solutions; a Mathematica visualization tool