A modular toolkit for bulk reconstruction

TL;DR

We address how boundary data in AdS/CFT can reveal bulk locality and causality by developing a modular-flow toolkit for heavy probes. The approach centers on conjectured rules that translate modular flow into boosted, piecewise geodesics, enabling extraction of the entangling surface and entanglement wedge structure from boundary correlators. Key results include a geometric interpretation of mirror operators via Δ^{1/2}, a framework for entanglement wedge nesting through double modular flow, and a boundary derivation of QNEC in the near-boundary limit. The work provides a practical, boundary-based method for bulk reconstruction and lays groundwork for connecting modular-flow dynamics to bulk Einstein equations and the quantum focusing condition.

Abstract

We introduce new tools for studying modular flow in AdS/CFT. These tools allow us to efficiently extract bulk information related to causality and locality. For example, we discuss the relation between analyticity in modular time and entanglement wedge nesting which can then be used to extract the location of the Ryu-Takayanagi (RT) surface directly from the boundary theory. Probing the RT surface close to the boundary our results reduce to the recent proof of the Quantum Null Energy Condition. We focus on heavy probe operators whose correlation functions are determined by spacelike geodesics. These geodesics interplay with the RT surface via a set of rules that we conjecture and give evidence for using the replica trick.

Figures (11)

• Figure 1: The first rule pertains to a configuration of operator insertions $(x,y)$ and entangling surface $m_A$ such that the smallest length configuration of spacelike geodesics that meet at some point $\xi$ on the entangling surface with a local relative boost of rapidity $2\pi s$. The modular flow correlator is then computable as a sum of lengths. The right figure is a zoom of the left figure showing only the transverse directions to $m_A$.
• Figure 2: The second rule pertains to a small deformation of the picture in \ref{['fig:rule1']}. In other words first order deformations in the parameters that lead to computable modular flow are still computable to second order in those deformations using a set of spacelike geodesics that meets on the actual entangling surface. The right figure shows an alternative description which is useful for calculations. Note that a choice to match in a related way along $\mathcal{H}^-$ would lead to the same result.
• Figure 3: Modular flowed geometry. The white regions are the two entanglement wedges that are the same as in the unflowed state. The blue region is effected by modular flow and in general unknown. In the right section we show some sample geodesics that compute correlation functions. The geodesic may enter the unknown region in which case we do not have control. If we manage to tune parameters so that this does not happen, we are left with a simple local analysis at the RT surface.
• Figure 4: Left: Correlation functions of operators with their mirrors, according to the rules, are computed via a geodesic that reflects of the entangling surface perpendicular to the surface. Right: this map is not one to one and there might be several points on the entangling surface that are represented by the same point on the boundary. This results in shadows, which are regions on the entangling surface that are not accessible with these mirror correlators. In the above example the shadow lies between the two reflection points. The above example is a cartoon and we have not worked through any real example, leaving this to future work.
• Figure 5: Double modular flow can also be computed if the parameters $x,y,s$ are tuned accordingly so that the geodesic threads itself through the various entanglement wedges.