This paper studies how a particular network of curves in empty anti-de Sitter (AdS) space can be used to describe the geometry inside a chosen subregion. The authors generalize a known idea — that all geodesics (shortest paths) in a highly symmetric space can be used to recover areas by counting intersections — to the case of an arbitrary connected subregion of vacuum AdS. They propose that the same web of curves, called partial-entanglement-entropy (PEE) threads, can be restricted to a subregion and still perfectly cover it.

PEE threads are bulk geodesics that are given a physical meaning by a boundary quantity called the two-point partial entanglement entropy (PEE). Unlike another earlier thread picture (bit threads), the PEE threads are fixed by the boundary entanglement data and they can cross each other. In prior work the PEE threads form an exact tessellation of Poincaré AdS. Here the authors analyze the kinematic space — the space of geodesics — for a general connected subregion a in vacuum AdS and identify the subset of geodesic segments that lie inside a. They show how to parameterize that subset and give a kinematic measure that counts intersections with bulk surfaces.

Using this construction, they show that the usual Ryu–Takayanagi (RT) surface for a boundary region can be recovered by finding the homologous surface inside the bulk that minimizes the number of intersections with the PEE network. That minimal intersection number equals the entanglement entropy from the RT formula. More generally, the paper uses the restricted PEE thread network to reconstruct bulk codimension-two surfaces by counting intersections, connecting the counting procedure to the Crofton formula from integral geometry (the mathematical statement that area equals an integral over intersections with geodesics).