What's the Point? Hole-ography in Poincare AdS

• Published in: arXiv.org 2018-02
• Main Authors: Espíndola, Ricardo, Guijosa, Alberto, Landetta, Alberto, Pedraza, Juan F
• Format: Article
• Language: English
• Subjects: Geodesy; Periodic variations; Reconstruction; Segments; Wedges
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1708.02958

In the context of the AdS/CFT correspondence, we study bulk reconstruction of the Poincare wedge of AdS\(_3\) via hole-ography, i.e., in terms of differential entropy of the dual CFT\(_2\). Previous work had considered the reconstruction of closed or open spacelike curves in global AdS, and of infinitely extended spacelike curves in Poincare AdS that are subject to a periodicity condition at infinity. Working first at constant time, we find that a closed curve in Poincare is described in the CFT by a family of intervals that covers the spatial axis at least twice. We also show how to reconstruct open curves, points and distances, and obtain a CFT action whose extremization leads to bulk points. We then generalize all of these results to the case of curves that vary in time, and discover that generic curves have segments that cannot be reconstructed using the standard hole-ographic construction. This happens because, for the nonreconstructible segments, the tangent geodesics fail to be fully contained within the Poincare wedge. We show that a previously discovered variant of the hole-ographic method allows us to overcome this challenge, by reorienting the geodesics touching the bulk curve to ensure that they all remain within the wedge. Our conclusion is that all spacelike curves in Poincare AdS can be completely reconstructed with CFT data, and each curve has in fact an infinite number of representations within the CFT.