Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Holographic representation of local bulk operators

Alex Hamilton, Daniel Kabat, Gilad Lifschytz, David A. Lowe

TL;DR

This work provides explicit constructions mapping local bulk AdS fields to non-local boundary operators via smeared boundary data, across global, Poincaré, and Rindler patches. It derives dimension- and patch-dependent smearing kernels, showing that boundary operators can have compact (or effectively compact in complexified boundary) support that scales with bulk radius. The authors analyze bulk locality, UV/IR implications, and finite-N holographic bounds, demonstrating how holography constrains the number and locality of commuting bulk operators. The results illuminate how AdS geometry, patch choice, and analytic continuation shape bulk operator reconstruction and highlight differences between even and odd dimensions, as well as the special role of Rindler coordinates in exposing scale-radius duality.

Abstract

The Lorentzian AdS/CFT correspondence implies a map between local operators in supergravity and non-local operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincare and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite N the number of independent commuting operators localized within a bulk volume saturates the holographic bound.

Holographic representation of local bulk operators

TL;DR

This work provides explicit constructions mapping local bulk AdS fields to non-local boundary operators via smeared boundary data, across global, Poincaré, and Rindler patches. It derives dimension- and patch-dependent smearing kernels, showing that boundary operators can have compact (or effectively compact in complexified boundary) support that scales with bulk radius. The authors analyze bulk locality, UV/IR implications, and finite-N holographic bounds, demonstrating how holography constrains the number and locality of commuting bulk operators. The results illuminate how AdS geometry, patch choice, and analytic continuation shape bulk operator reconstruction and highlight differences between even and odd dimensions, as well as the special role of Rindler coordinates in exposing scale-radius duality.

Abstract

The Lorentzian AdS/CFT correspondence implies a map between local operators in supergravity and non-local operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincare and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite N the number of independent commuting operators localized within a bulk volume saturates the holographic bound.

Paper Structure

This paper contains 17 sections, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Global AdS
  3. Preliminaries
  4. Even AdS: mode sum approach
  5. Odd AdS: mode sum approach
  6. Poincaré smearing
  7. Even AdS
  8. Odd AdS
  9. Rindler smearing in AdS${}_3$
  10. Massless field in Rindler coordinates
  11. Wick rotating to de Sitter space
  12. Physical consequences
  13. Bulk locality and UV/IR
  14. Finite $N$ and the holographic bound
  15. Even AdS: global Greens function
  16. ...and 2 more sections

Figures (3)

  • Figure 1: In global coordinates AdS resembles an infinite cylinder. We've drawn the light cones emanating from a bulk point and intersecting the boundary. The CFT operator has support on the strip indicated in yellow, at spacelike separation from the bulk point.
  • Figure 2: Branch cuts in the $\hat{t}'$ plane are located at ${\rm Im} \, \hat{t}' = \pm \pi / 2$.
  • Figure 3: For $\cosh (\hat{t} - \hat{t}') > 1/\sqrt{1 - r_+^2/r^2}$ the branch cuts in the $\hat{\phi}'$ plane are at ${\rm Im} \, \hat{\phi}' = \pm \pi / 2$ (left panel). When $\cosh(\hat{t} - \hat{t}') = 1/\sqrt{1 - r_+^2/r^2}$ four of the branch points touch, and for $\cosh(\hat{t} - \hat{t}') < 1/\sqrt{1 - r_+^2/r^2}$ the branch cuts are cross-shaped (right panel).