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

TL;DR

The paper investigates how bulk locality is encoded in boundary data within AdS/CFT, focusing on subregion dualities. It develops a reconstruction map from boundary one-point functions to bulk fields, and introduces a geometric null-geodesic criterion to gauge when continuous classical reconstruction is possible. The authors demonstrate continuous reconstruction in global AdS but show failure for AdS-Rindler with local boundary data, arguing that nonlocal boundary operators are essential for subregion dualities. They extend the analysis to Poincare and Milne patches, relate reconstruction to RG flow, and discuss implications for black hole geometries, ultimately supporting a nuanced view where subregion duality may exist in a nonlocal CFT sense. The work highlights how null geodesics govern reconstructibility and points toward nonlocal CFT structures as key to encoding bulk subregions.

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.

Figures (6)

• Figure 1: Here we show the AdS-Rindler wedge inside of global AdS, which can be defined as the intersection of the past of point $A$ with the future of point $B$. The asymptotic boundary is the small causal diamond defined by points $A$ and $B$. The past lightcone of $A$ and the future lightcone of $B$ intersect along the dashed line, which is a codimension-2 hyperboloid in the bulk. There is a second AdS-Rindler wedge, defined by the points antipodal to $A$ and $B$, that is bounded by the same hyperboloid in the bulk. We refer to such a pair as the "right" and "left" AdS-Rindler wedges.
• Figure 2: Here we depict Poincare-Milne space, together with an AdS-Rindler space that it contains. The bulk of Poincare-Milne can be defined as the intersection of the past of point $A$ with the future of line $BE$. Clearly this region contains the AdS-Rindler space which is the intersection of the past of $A$ and the future of $B$. Furthermore, the asymptotic boundary of the Poincare-Milne space and the AdS-Rindler space is identical, being the causal diamond defined by $A$ and $B$ on the boundary.
• Figure 3: This is one of many null geodesics which passes through AdS-Rindler space without reaching the AdS-Rindler boundary. The four highlighted points on the trajectory are (bottom to top) its starting point on the near side of the global boundary, its intersection with the past Rindler horizon, its intersection with the future Rindler horizon, and its endpoint on the far side of the global boundary.
• Figure 4: On the left we show the effective potential for a null geodesic in a spherical black hole, and on the right the same for a planar black brane. In the case of a spherical black hole, there is a potential barrier which traps some null geodesics in the $r< 3G_NM$ region. Therefore continuous reconstruction from the boundary is not possible for the region $r<3G_NM$. In the planar case, there are null geodesics reach arbitrarily large finite $r$ without making it to the boundary. Hence there is no bulk region which can be continuously reconstructed from the boundary data.
• Figure 5: The geometry defining the Hartle-Hawking state for AdS-Rindler. Half of the Lorentzian geometry, containing the $t>0$ portion of both the left and right AdS-Rindler spaces, is glued to half of the Euclidean geometry. The left and right sides are linked by the Euclidean geometry, and the result is that the state at $t=0$ is entangled between the two halves.