Null Geodesics, Local CFT Operators and AdS/CFT for Subregions

TL;DR

This paper explores the limitations of reconstructing bulk regions in AdS/CFT, showing that certain subregions like AdS-Rindler cannot be reconstructed solely from local boundary data, implying the necessity of nonlocal operators.

Contribution

It demonstrates the failure of continuous bulk reconstruction for AdS-Rindler subregions from local CFT data and proposes a geometric criterion involving null geodesics for when subregion duality is possible.

Findings

Global AdS/CFT reduces to boundary value problem with local operators.

AdS-Rindler subregion cannot be reconstructed from local boundary data.

Null geodesic endpoint criterion for subregion duality.

Abstract

We investigate the nature of the AdS/CFT duality between a subregion of the bulk and its boundary. In global AdS/CFT in the classical G_N=0 limit, the duality reduces to a boundary value problem that can be solved by restricting to one-point functions of local operators in the CFT. We show that the solution of this boundary value problem depends continuously on the CFT data. In contrast, the AdS-Rindler subregion cannot be continuously reconstructed from local CFT data restricted to the associated boundary region. Motivated by related results in the mathematics literature, we posit that a continuous bulk reconstruction is only possible when every null geodesic in a given bulk subregion has an endpoint on the associated boundary subregion. This suggests that a subregion duality for AdS-Rindler, if it exists, must involve nonlocal CFT operators in an essential way.