Holographic definition of points and distances

Abstract

We discuss the way in which field theory quantities assemble the spatial geometry of three-dimensional anti--de Sitter space (${\mathrm{AdS}}_{3}$). The field theory ingredients are the entanglement entropies of boundary intervals. A point in ${\mathrm{AdS}}_{3}$ corresponds to a collection of boundary intervals which is selected by a variational principle we discuss. Coordinates in ${\mathrm{AdS}}_{3}$ are integration constants of the resulting equation of motion. We propose a distance function for this collection of points, which obeys the triangle inequality as a consequence of the strong subadditivity of entropy. Our construction correctly reproduces the static slice of ${\mathrm{AdS}}_{3}$ and the Ryu-Takayanagi relation between geodesics and entanglement entropies. We discuss how these results extend to quotients of ${\mathrm{AdS}}_{3}$---the conical defect and the BTZ geometries. In these cases, the set of entanglement entropies must be supplemented by other field theory quantities, which can carry the information about lengths of nonminimal geodesics.