Local bulk operators in AdS/CFT and the fate of the BTZ singularity

TL;DR

The paper reviews how local bulk operators in AdS can be represented by nonlocal boundary operators and how bulk locality emerges in the semiclassical, large-N limit, while signaling breakdown at finite N. It extends the discussion to BTZ black holes, showing how horizons and singularities appear in the large-N limit via boundary correlators and smearing constructions. At finite N, the authors propose a modified smearing approach with a cutoff that effectively averages over microstates, suggesting the BTZ interior describes an ensemble rather than a single microstate and that quantum gravity resolves the singularity. Together, these results argue that bulk locality is an emergent, approximate feature that requires microstate averaging to remain valid inside horizons at finite N.

Abstract

This paper has two parts. First we review the description of local bulk operators in Lorentzian AdS in terms of non-local operators in the boundary CFT. We discuss how bulk locality arises in pure AdS backgrounds and how it is modified at finite N. Next we present some new results on BTZ black holes: local operators can be defined inside the horizon of a finite N BTZ black hole, in a way that suggests the BTZ geometry describes an average over black hole microstates, but with finite N effects resolving the singularity.

Figures (4)

• Figure 1: The field at a bulk point in de Sitter space can be expressed in terms of data on the past de Sitter boundary. The slice $Y = 0$ also describes a region in AdS. So we can also regard this as expressing the field in AdS in terms of data on the complexified AdS boundary.
• Figure 2: Smearing functions for two bulk points separated only in the $Z$ direction. The smearing functions overlap on the boundary, nonetheless the smeared operators commute at infinite $N$.
• Figure 3: The smearing functions have support on the two jagged lines. For $\vert \Delta X \vert > Z$ they are spacelike separated.
• Figure 4: Penrose diagram of the $(t,\phi)$ plane. The support of $\widetilde{K}$ is indicated by the jagged line. Points in the shaded region are spacelike separated from the support of $\widetilde{K}$. When the smearing function extends from $-\hat{t}_{\rm max}$ to $+\hat{t}_{\rm max}$ the shaded region is characterized by $\vert \hat{\phi} \vert > \hat{t}_{\rm max} + \vert \hat{t} \vert$.