HKLL bulk reconstruction for small ∆

Abstract

We discuss the extension of the HKLL (Hamilton, Kabat, Lifschytz, and Lowe) bulk reconstruction for non-interacting scalar fields corresponding to conformal weights ∆ smaller than the original condition ∆ > d − 1. We give explicit formulas for the cases d − 2 < ∆ ≤ d − 1 and ∆ = d − s with integer s. In the latter case we show that smearing CFT fields over a region of the boundary consisting of points light-like separated from the bulk point is sufficient for bulk reconstruction, whereas in general smearing over all light-like and space-like separated points is required.