Dressing bulk fields in AdS${}_3$

TL;DR

This paper develops a method to reconstruct charged bulk scalar fields in AdS${}_3$ from CFT operators, accounting for interactions with conserved currents, and explicitly computes the resulting bulk correlators.

Contribution

It introduces a systematic way to dress bulk fields in AdS${}_3$ using a tower of smeared operators, enabling direct calculation of bulk correlators from the CFT.

Findings

Bulk correlators are expressed as polynomials of kinematic invariants.

The method captures the dressing of bulk fields by a generalized Wilson line.

The approach is valid to order 1/N in the large N expansion.

Abstract

We study a set of CFT operators suitable for reconstructing a charged bulk scalar field $\phi$ in AdS${}_3$ (dual to an operator ${\cal O}$ of dimension $\Delta$ in the CFT) in the presence of a conserved spin-$n$ current in the CFT. One has to sum a tower of smeared non-primary scalars $\partial_{+}^{m} J^{(m)}$, where $J^{(m)}$ are primaries with twist $\Delta$ and spin $m$ built from ${\cal O}$ and the current. The coefficients of these operators can be fixed by demanding that bulk correlators are well-defined: with a simple ansatz this requirement allows us to calculate bulk correlators directly from the CFT. They are built from specific polynomials of the kinematic invariants up to a freedom to make field redefinitions. To order $1/N$ this procedure captures the dressing of the bulk scalar field by a radial generalized Wilson line.