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Bulk reconstruction using timelike entanglement in (A)dS

Avijit Das, Shivrat Sachdeva, Debajyoti Sarkar

TL;DR

The paper develops a timelike entanglement framework in 2D CFTs dual to AdS spacetimes, introducing timelike EE $S_A^{(T)}$ and timelike modular Hamiltonians $H_A^{(T)}$ as boundary data to reconstruct bulk fields beyond the RT wedge. By translating boundary timelike constraints into bulk HKLL-type smearing, it obtains explicit scalar reconstruction in AdS3 Poincaré, outside and inside BTZ, and in de Sitter flat slicing, without requiring bulk dynamics. A key result is that timelike modular flows can access regions behind horizons, enabling subhorizon holography through boundary data alone and revealing a consistent geometric picture via extremal surfaces that extend into the bulk. These findings illuminate potential implications for black hole information and cosmological horizons and suggest promising extensions to higher dimensions and more general holographic states.

Abstract

It is well-known that the entanglement entropies for spacelike subregions, and the associated modular Hamiltonians play a crucial role in the bulk reconstruction program within Anti de-Sitter (AdS) holography. Explicit examples of HKLL map exist mostly for the cases where the emergent bulk region is the so-called entanglement wedge of the given boundary subregion. However, motivated from the complex pseudo-entropy in Euclidean conformal field theories (CFT), one can talk about a `timelike entanglement' in Lorentzian CFTs dual to AdS spacetimes. One can then utilize this boundary timelike entanglement to define a boundary `timelike modular Hamiltonian'. We use constraints involving these Hamiltonians in a manner similar to how it was used for spacelike cases, and write down bulk operators in regions which are not probed by an RT surface corresponding to a single CFT. In the context of two dimensional CFT, we re-derive the HKLL formulas for free bulk scalar fields in three examples: in AdS Poincaré patch, inside and outside of the AdS black hole, and for de Sitter flat slicings. In this method, one no longer requires the knowledge of bulk dynamics for sub-horizon holography.

Bulk reconstruction using timelike entanglement in (A)dS

TL;DR

The paper develops a timelike entanglement framework in 2D CFTs dual to AdS spacetimes, introducing timelike EE and timelike modular Hamiltonians as boundary data to reconstruct bulk fields beyond the RT wedge. By translating boundary timelike constraints into bulk HKLL-type smearing, it obtains explicit scalar reconstruction in AdS3 Poincaré, outside and inside BTZ, and in de Sitter flat slicing, without requiring bulk dynamics. A key result is that timelike modular flows can access regions behind horizons, enabling subhorizon holography through boundary data alone and revealing a consistent geometric picture via extremal surfaces that extend into the bulk. These findings illuminate potential implications for black hole information and cosmological horizons and suggest promising extensions to higher dimensions and more general holographic states.

Abstract

It is well-known that the entanglement entropies for spacelike subregions, and the associated modular Hamiltonians play a crucial role in the bulk reconstruction program within Anti de-Sitter (AdS) holography. Explicit examples of HKLL map exist mostly for the cases where the emergent bulk region is the so-called entanglement wedge of the given boundary subregion. However, motivated from the complex pseudo-entropy in Euclidean conformal field theories (CFT), one can talk about a `timelike entanglement' in Lorentzian CFTs dual to AdS spacetimes. One can then utilize this boundary timelike entanglement to define a boundary `timelike modular Hamiltonian'. We use constraints involving these Hamiltonians in a manner similar to how it was used for spacelike cases, and write down bulk operators in regions which are not probed by an RT surface corresponding to a single CFT. In the context of two dimensional CFT, we re-derive the HKLL formulas for free bulk scalar fields in three examples: in AdS Poincaré patch, inside and outside of the AdS black hole, and for de Sitter flat slicings. In this method, one no longer requires the knowledge of bulk dynamics for sub-horizon holography.
Paper Structure (35 sections, 200 equations, 13 figures)

This paper contains 35 sections, 200 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Timelike entanglement entropy
  4. CFT methods
  5. Rotating the interval
  6. Wick rotating the coordinates
  7. Holographic methods
  8. Pure AdS$_3$
  9. BTZ Blackhole
  10. Timelike modular Hamiltonian
  11. Timelike interval in CFT$_2$ on a periodic space
  12. Timelike interval in CFT$_2$ at finite temperature
  13. Revisiting AdS$_3$ Poincaré reconstruction
  14. AdS$_3$ Poincaré reconstruction using $\tilde{H}^{(T)}$
  15. Outside BTZ horizon
  16. ...and 20 more sections

Figures (13)

  • Figure 1:
  • Figure 2: Timelike entanglement entropy via a Wick rotation.
  • Figure 3: The union of the spacelike (green) and timelike (red) surfaces constitutes the extremal surface for the (blue) timelike subregion $A$.
  • Figure 4: The (left) right panel surfaces respectively depict the extremal surfaces for a timelike subregion which is (not) symmetric around boundary $t=0$.
  • Figure 5: Two non-intersecting timelike subsystems $A$ and $B$ whose spacelike parts of the corresponding extremal surfaces intersect at a bulk point $P$.
  • ...and 8 more figures