HKLL for the non-normalizable mode

We discuss various aspects of HKLL bulk reconstruction for the free scalar field in AdSd+1. First, we consider the spacelike reconstruction kernel for the non-normalizable mode in global coordinates. We construct it as a mode sum. In even bulk dimensions, this can be reproduced using a chordal Green's function approach that we propose. This puts the global AdS results for the non-normalizable mode on an equal footing with results in the literature for the normalizable mode. In Poincar & eacute; AdS, we present explicit mode sum results in general even and odd dimensions for both normalizable and non-normalizable kernels. For generic scaling dimension delta, these can be re-written in a form that matches with the global AdS results via an antipodal mapping, plus a remainder. We are not aware of a general argument in the literature for dropping these remainder terms, but we note that a slight complexification of a boundary spatial coordinate (which we call an ic prescription) allows us to do so in cases where delta is (half-) integer. Since the non-normalizable mode turns on a source in the CFT, our primary motivation for considering it is as a step towards understanding linear wave equations in general spacetimes from a holographic perspective. But when the scaling dimension delta is in the Breitenlohner-Freedman window, we note that the construction has some interesting features within AdS/CFT.

Automated summary

This paper discusses various aspects of the HKLL bulk reconstruction for the free scalar field in AdSd+1. The authors consider the spacelike reconstruction kernel for the non-normalizable mode in global coordinates, constructing it as a mode sum. They propose a chordal Green's function approach to reproduce it in even bulk dimensions, putting the global AdS results for the non-normalizable mode on par with results for the normalizable mode in the literature. Explicit mode sum results in Poincaré AdS are presented for both normalizable and non-normalizable kernels in general even and odd dimensions. These results can be rewritten to match the global AdS results through an antipodal mapping and a remainder for generic scaling dimension delta. The construction of the non-normalizable mode is motivated by understanding linear wave equations in general spacetimes from a holographic perspective. The authors note interesting features within AdS/CFT when the scaling dimension delta is in the Breitenlohner-Freedman window.