CFT representation of interacting bulk gauge fields in AdS

Abstract

We develop the representation of interacting bulk gauge fields and charged scalar matter in anti--de Sitter in terms of nonlocal observables in the dual conformal field theory (CFT). We work in the holographic gauge in the bulk, ${A}_{z}=0$. The correct statement of micro-causality in the holographic gauge is somewhat subtle, so we first discuss it from the bulk point of view. We then show that in the $1/N$ expansion, CFT correlators can be lifted to obtain bulk correlation functions that satisfy microcausality. This requires adding an infinite tower of higher-dimension multitrace operators to the CFT definition of a bulk observable. For conserved currents, the Ward identities in the CFT prevent the construction of truly local bulk operators (i.e., operators that commute at spacelike separation with everything); however, the resulting nonlocal commutators are exactly those required by the bulk Gauss constraint. In contrast, a CFT which only has nonconserved currents can be lifted to a bulk theory which is truly local. Although our explicit calculations are for gauge theory, similar statements should hold for gravity.