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