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A toy model of black hole complementarity

Souvik Banerjee, Jan-Willem Bryan, Kyriakos Papadodimas, Suvrat Raju

TL;DR

The paper studies a holographic setup where a short boundary time band yields a bulk central diamond causally disconnected from the boundary. It shows that a small, simple-algebra of boundary operators satisfies a Reeh-Schlieder-type property, allowing approximation of low-energy bulk states, while exact vacuum annihilation is forbidden; interior bulk operators (precursors) can be reconstructed from highly composite boundary polynomials using an approximate vacuum projector. This explicit precursor construction demonstrates how interior information can be encoded nonlocally in boundary data, offering a concrete toy realization of black hole complementarity and the emergence of bulk locality from boundary theory. The work clarifies the role of operator complexity and time-band restrictions in AdS/CFT and motivates further connections to modular structure and entanglement-based reconstruction.

Abstract

We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called "precursors". We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.

A toy model of black hole complementarity

TL;DR

The paper studies a holographic setup where a short boundary time band yields a bulk central diamond causally disconnected from the boundary. It shows that a small, simple-algebra of boundary operators satisfies a Reeh-Schlieder-type property, allowing approximation of low-energy bulk states, while exact vacuum annihilation is forbidden; interior bulk operators (precursors) can be reconstructed from highly composite boundary polynomials using an approximate vacuum projector. This explicit precursor construction demonstrates how interior information can be encoded nonlocally in boundary data, offering a concrete toy realization of black hole complementarity and the emergence of bulk locality from boundary theory. The work clarifies the role of operator complexity and time-band restrictions in AdS/CFT and motivates further connections to modular structure and entanglement-based reconstruction.

Abstract

We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called "precursors". We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.

Paper Structure

This paper contains 8 sections, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The setup
  3. Operator algebras in time bands
  4. Explicit Construction of Arbitrary States Using Operators in the Band
  5. Interior operators and precursors
  6. Projector on the Vacuum
  7. Construction of ${\cal O}_{n, \ell}$
  8. Conclusion

Figures (2)

  • Figure 1: Degrees of freedom at $x_{\rm in}$ are identified with complicated combinations of those at $x_{\rm out}$.
  • Figure 2: The $\text{time band}$${\cal B}$ of length $T<\pi$ on the boundary of AdS spacetime, the diamond shaped region $\cal D$ in the bulk and its causal complement, the annular region $\overline{\cal D}$.