Integral Geometry and Holography

TL;DR

This work introduces kinematic space, a Lorentzian auxiliary geometry defined by the Crofton form and conditional mutual information, to translate boundary entanglement data into bulk geometric quantities in the AdS3/CFT2 setting. By identifying the Crofton density with I(A,C|B) and exploiting strong subadditivity, it shows that lengths of bulk curves correspond to volumes in kinematic space, enabling bulk reconstruction from boundary entanglement entropies. The framework unifies integral geometry with holographic entanglement and reveals connections to differential entropy, enabling a boundary-driven dictionary and suggesting deep links to tensor networks like MERA and sub-AdS locality. The canonical AdS3 example demonstrates how kinematic space becomes dS2 for the static slice, and the discussion points to broader implications for dS/CFT, holographic dualities, and information-geometric perspectives on spacetime emergence.

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.

Figures (23)

• Figure 1: Kinematic space, $K_{\mathcal{M}}$, acts as an intermediary that translates between the boundary language of information theory describing the state $\psi$ and the bulk language of geometry for the manifold $\mathcal{M}$.
• Figure 2: The notation of Sec. \ref{['flatplane']}.
• Figure 3: Left: A nonconvex curve $\gamma$ in flat space. Some lines intersect the curve twice, while others intersect four times. Right: The region of kinematic space showing the set of curves intersecting $\gamma$. The black curves correspond to geodesics tangent to $\gamma$, and they form the boundary of the region of intersecting curves. The light-shaded region corresponds to geodesics intersecting twice, while the dark-shaded region corresponds to geodesics intersecting four times. This intersection number enters the Crofton formula (\ref{['flatfinalcrofton']}).
• Figure 4: The parameterization of kinematic space for the hyperbolic plane. We denote the opening angle of a geodesic by $\alpha$ and the angular coordinate of the center of the geodesic by $\theta$. The endpoints are labeled $u$ and $v$.
• Figure 5: We characterize a convex curve $\gamma$ with a function $v(u)$ chosen so that the geodesic from $u$ to $v(u)$ is tangent to $\gamma$. When $\gamma$ is not differentiable, the geodesic meets $\gamma$ at an isolated point.