Holographic representation of bulk fields with spin in AdS/CFT

TL;DR

This paper shows how bulk fields with spin, including gauge fields and gravitons, can be represented as non-local CFT observables via explicit smearing functions in holographic gauge. It demonstrates that while bulk gauge fields and metric perturbations exhibit non-local commutators, the gauge-invariant content—field strengths and the Weyl tensor—satisfies bulk causality. For massive vectors, locality is restored despite a non-conserved boundary current, by mixing in higher-dimension operators; for massless gauge fields and gravity, causality is encoded in curvature-like quantities. The work also discusses extending these constructions to interacting theories and general backgrounds, proposing a path toward master bulk operators defined purely within the CFT framework under suitable large-N conditions.

Abstract

We develop the representation of bulk fields with spin one and spin two in anti-de Sitter space, as non-local observables in the dual CFT. Working in holographic gauge in the bulk, at leading order in 1/N bulk gauge fields are obtained by smearing boundary currents over a sphere on the complexified boundary, while linearized metric fluctuations are obtained by smearing the boundary stress tensor over a ball. This representation respects AdS covariance up to a compensating gauge transformation. We also consider massive vector fields, where the bulk field is obtained by smearing a non-conserved current. We compute bulk two-point functions and show that bulk locality is respected. We show how to include interactions of massive vectors using 1/N perturbation theory, and we comment on the issue of general backgrounds.

Figures (1)

• Figure 1: Integration contour for $I_n$. At large spacelike separation the pole is far up the imaginary axis. The pole moves down and crosses the integration contour when $x = z$; one can continue to smaller values of $x$ by deforming the contour. The integral may be singular when $x \rightarrow 0^+$ and the pole moves to $-i \infty$. There are singularities when $x \rightarrow \pm i z$ and the pole hits an endpoint of the integration contour.