State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

TL;DR

The paper proposes and tests a state-dependent bulk-boundary mapping in AdS/CFT that doubles boundary operators to construct mirror operators behind black hole horizons. By enforcing that these tilde operators reproduce correct thermal behavior inside low-point correlators, the authors achieve an effectively local bulk EFT behind the horizon while preserving unitarity and avoiding firewall-like paradoxes. They provide explicit constructions, toy models (harmonic oscillators, spin chains), and a comprehensive treatment of non-Abelian charges, non-equilibrium states, and connections to Tomita-Takesaki modular theory. The work argues that horizon structure and complementarity can coexist with unitary evolution, clarifying issues like strong subadditivity, left inverses, and small corrections, and suggests a deep mathematical underpinning via modular theory. Overall, the approach offers a coherent framework for interior bulk physics in AdS/CFT that remains consistent with boundary dynamics and quantum information constraints, while highlighting its state-dependent nature and limits of applicability in high-point correlators.

Abstract

We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in arxiv:1304.6483 and arxiv:1307.4706, in favour of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the "frozen vacuum" objections of arxiv:1308.3697. Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.

Figures (12)

• Figure 1: A Black Hole is created in AdS by injecting matter from the boundary. We are interested in the red-colored patch $P$, behind the horizon, which is far away from both the infalling matter and the singularity
• Figure 2: A toy model of a black hole (small black-circle in the center) emitting Hawking radiation. We are quantizing a massless field between the two solid circles, one of which is a Page distance away. The emission is mostly in s-waves if the inner cutoff is far enough from the horizon.
• Figure 3: Behaviour of the entanglement entropy $S_n$ with $n$.
• Figure 4: Modeling Hawking Radiation in a "Localized Black Hole": Hawking Quanta $R$ emitted towards the boundary reflect and fall back, but quanta emitted in the non-compact directions escape to infinity.
• Figure 5: $\widetilde{b}^{\dagger}$ is a sparse operator, and it maps the intersection of ${\cal H}_{\Psi}$ with the space of states of average energy $E$ to the intersection of ${\cal H}_{\Psi}$ with the states of average energy $E - \omega_n$. The precise domain and range depend on the base state $|\Psi \rangle$.