Nuts and Bolts for Creating Space

TL;DR

This paper explores how entanglement entropies in boundary field theories encode the geometry of AdS3 space, proposing a new distance function consistent with holographic principles and extending to quotient geometries.

Contribution

It introduces a novel method to reconstruct AdS3 geometry from boundary entanglement entropies and proposes a distance function satisfying the triangle inequality.

Findings

Reproduces static AdS3 slice from entanglement data

Proposes a distance function obeying triangle inequality

Extends framework to conical defect and BTZ geometries

Abstract

We discuss the way in which field theory quantities assemble the spatial geometry of three-dimensional anti-de Sitter space (AdS3). The field theory ingredients are the entanglement entropies of boundary intervals. A point in AdS3 corresponds to a collection of boundary intervals, which is selected by a variational principle we discuss. Coordinates in AdS3 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 AdS3 and the Ryu-Takayanagi relation between geodesics and entanglement entropies. We discuss how these results extend to quotients of AdS3 -- 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 non-minimal geodesics.