Toward stringy horizons

TL;DR

The paper develops a boundary-algebra framework to characterize stringy bulk horizons and quantum extremal surfaces, introducing a causal depth parameter $\mathcal{T}$ and half-sided modular inclusions as diagnostic tools. It shows that above the Hawking–Page temperature, the large $N$ limit of ${\mathcal N}=4$ SYM at finite $\lambda$ exhibits an emergent stringy horizon, while certain toy models do not, and it builds a formalism (ER algebra, modular depth) to discuss connectivity and entropy in the stringy regime. It also connects the horizon/RT/QES questions to spectral data via the exponential type of the boundary spectral function, and discusses potential violations of the bulk equivalence principle, as well as extensions of the ER=EPR program to stringy settings and evaporating black holes. The work points toward a rigorous, algebraic route to stringy bulk geometry, with implications for nonlocality, reconstruction, and the thermodynamics of holographic systems beyond Einstein gravity.

Abstract

We take a first step towards developing a new language to describe causal structure, event horizons, and quantum extremal surfaces (QES) for the bulk description of holographic systems beyond the standard Einstein gravity regime. By considering the structure of boundary operator algebras, we introduce a stringy ``causal depth parameter'', which quantifies the depth of the emergent radial direction in the bulk, and a certain notion of ergodicity on the boundary. We define stringy event horizons in terms of the half-sided inclusion property, which is related to a stronger notion of boundary ergodic or quantum chaotic behavior. Using our definition, we argue that above the Hawking--Page temperature, there is an emergent sharp horizon structure in the large $N$ limit of $\mathcal{N}=4$ Super-Yang--Mills at finite nonzero 't Hooft coupling. In contrast, some previously considered toy models of black hole information loss do not have a stringy horizon. Our methods can also be used to probe violations of the equivalence principle for the bulk gravitational system, and to explore aspects of stringy nonlocality.

Figures (8)

• Figure 1.1: A simple illustration of subregion/subalgebra duality in the vacuum state of strongly coupled $\mathcal{N}=4$ Super-Yang--Mills theory. The region ${\mathfrak{a}}$, a spherical Rindler region in the bulk, is dual to the algebra $\mathcal{S}_I$ of large $N$ boundary observables in the time band $I=(-t_0,t_0)$. The bulk diamond ${\mathfrak{b}}$, which does not touch the boundary, can be identified with the commutant of the algebra $\mathcal{S}_I$.
• Figure 2.1: Diagnosing the presence of a bifurcate horizon from the algebraic structure of boundary time bands.
• Figure 2.2: A heuristic picture of the depth parameter in thermal AdS. In this case, there is a point, depicted in red, which sits at the "center" of the bulk. The depth parameter $\mathcal{T}$ can be seen as the boundary time interval that can be reached by shooting light rays from this point. Here, $\mathcal{T}=\pi$.
• Figure 2.3: Half-sided inclusions for future and past horizons. In the thermofield double state at strong coupling and high temperature, the algebras $\mathcal{N}^+$ and $\mathcal{N}^-$, which supported on future and past semi-infinite time intervals, are dual to the red and blue wedges in the bulk, respectively. In particular, it is the fact that they are inequivalent to the full algebra of right boundary observables that allows for the emergence of nontrivial future and past half-sided modular inclusions, which we will promote to a definition of a horizon in the TFD state in the stringy regime.
• Figure 2.4: Cartoons for various types of two-sided states. These examples also illustrate that the causal depth parameter does not allow to predict spacetime connectivity in a general state. The cases (a) and (b) both have infinite causal depth parameter, but (a) is connected whereas (b) is disconnected. Similarly, (b) and (c) are both disconnected but (b) has infinite causal depth parameter while (c) has finite causal depth parameter. Finally case (d) is disconnected but has a future horizon.

Theorems & Definitions (47)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9
• proof