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Local bulk operators in AdS/CFT: a boundary view of horizons and locality

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

TL;DR

The paper develops a boundary representation for local bulk fields in AdS/CFT by constructing non-local boundary operators with smearing kernels that map bulk observables to boundary data, focusing on AdS${}_2$ as a tractable example. It demonstrates how bulk locality emerges from spacelike-separated boundary support and shows how to describe bulk insertions behind horizons, including inside the Rindler horizon of AdS black holes, by boundary operators that act on one or both copies of the thermofield-double CFT. The results provide a concrete mechanism to recover bulk correlators from CFT correlators and explain bulk light-cone singularities as UV boundary phenomena, while outlining generalizations to higher-dimensional AdS black holes. The framework offers new tools to probe horizons, locality, and black hole interiors within the AdS/CFT correspondence, with potential extensions beyond the semiclassical regime.

Abstract

We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS_2 as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincare patch and inside the Rindler horizon of a two dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

Local bulk operators in AdS/CFT: a boundary view of horizons and locality

TL;DR

The paper develops a boundary representation for local bulk fields in AdS/CFT by constructing non-local boundary operators with smearing kernels that map bulk observables to boundary data, focusing on AdS as a tractable example. It demonstrates how bulk locality emerges from spacelike-separated boundary support and shows how to describe bulk insertions behind horizons, including inside the Rindler horizon of AdS black holes, by boundary operators that act on one or both copies of the thermofield-double CFT. The results provide a concrete mechanism to recover bulk correlators from CFT correlators and explain bulk light-cone singularities as UV boundary phenomena, while outlining generalizations to higher-dimensional AdS black holes. The framework offers new tools to probe horizons, locality, and black hole interiors within the AdS/CFT correspondence, with potential extensions beyond the semiclassical regime.

Abstract

We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS_2 as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincare patch and inside the Rindler horizon of a two dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

Paper Structure

This paper contains 17 sections, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Global smearing functions
  3. AdS${}_2$ generalities
  4. Green's function approach
  5. Mode sum approach
  6. Poincaré coordinates
  7. Smearing functions
  8. Going behind the Poincaré horizon
  9. Recovering bulk correlators
  10. Light-cone singularities in the bulk
  11. AdS${}_2$ black holes
  12. Smearing functions
  13. Thermofield interpretation and black holes
  14. Higher dimensional black holes
  15. Conclusions
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Global smearing function for a $\Delta = 3$ bulk operator located at $\rho = 0$.
  • Figure 2: Smearing function for a bulk operator with integer $\Delta$ located outside the Poincaré patch.
  • Figure 3: Smearing function for an integer-dimension bulk operator located behind the Rindler horizon.
  • Figure 4: The Kruskal patch, indicated on the AdS Penrose diagram (left) and in the $uv$ plane (right).