Bulk reconstruction for anti-symmetric and symmetric gauge fields: $p$-forms and the graviton

Abstract: We consider the HKLL bulk reconstruction procedure for $p-$form fields and graviton in empty AdS${d + 1}$. We derive spacelike bulk reconstruction kernels for the $p$-forms and the graviton in the Poinarc\'e patch of AdS${d+1}$. The kernels are first derived via a mode-sum approach in arbitrary even dimensions. The appropriate AdS-covariant fields are identified and the corresponding kernels obtained via the mode sum approach for these fields. We present arguments for casting these kernels in terms of the AdS chordal distance. Introducing an antipodal-like mapping, the kernels are cast in a spacelike form. An alternative derivation is presented for these kernels, by using a chordal Green's function approach. From the asymptotic expansion of the bulk fields and using Green's theorem, we determine the spacelike kernels for both mode of $p-$forms and the graviton, which are show to agree with the kernels obtained by the mode-sum approach. The massive $p-$form fields, as well as the relevant Brietenlohner-Freedman bounds are also discussed.