CFT representation of interacting bulk gauge fields in AdS

TL;DR

This paper constructs a holographic representation of interacting gauge fields and charged scalars in AdS using non-local CFT observables, addressing micro-causality and gauge constraints.

Contribution

It demonstrates how to lift CFT correlators to bulk AdS correlators satisfying micro-causality by adding multi-trace operators, and discusses implications for gauge and gravity theories.

Findings

Bulk correlators can be reconstructed from CFT with multi-trace operators.

Conserved currents impose non-locality consistent with gauge constraints.

Non-conserved currents allow truly local bulk operators.

Abstract

We develop the representation of interacting bulk gauge fields and charged scalar matter in AdS in terms of non-local observables in the dual CFT. We work in holographic gauge in the bulk, A_z = 0. The correct statement of micro-causality in 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 which satisfy micro-causality. This requires adding an infinite tower of higher-dimension multi-trace 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 non-local commutators are exactly those required by the bulk Gauss constraint. In contrast a CFT which only has non-conserved 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.