Notes on the ambient approach to boundary values of AdS gauge fields

TL;DR

The authors develop an ambient-space formulation with $O(d,2)$ symmetry to analyze boundary values of AdS$_{d+1}$ gauge fields and relate them to $d$-dimensional conformal currents and shadow fields. They then extend this by a parent formulation that lifts ambient constructions to intrinsic AdS or conformal spaces, enabling an explicit AdS/boundary link and a covariant, unfolded-like treatment. For totally symmetric fields such as the Fronsdal fields, the near-boundary data split into currents and shadows in general, with a striking parity-dependent exception: in odd boundary dimension the non-normalizable sector simultaneously encodes a Fradkin–Tseytlin field along with the shadow/current structure. The framework accommodates scalar and higher-spin cases, clarifies the role of the Fradkin–Tseytlin equations, and provides a blueprint for extending to nonlinear higher-spin boundary dynamics via BRST/AKSZ and unfolded techniques.

Abstract

The ambient space of dimension d+2 allows to formulate both fields on AdS(d+1) and conformal fields in d dimensions such that the symmetry algebra o(d,2) is realized linearly. We elaborate an ambient approach to the boundary analysis of gauge fields on anti de Sitter spacetime. More technically, we use its parent extension where fields are still defined on AdS or conformal space through arbitrary intrinsic coordinates while the ambient construction works in the target space. In this way, a manifestly local and o(d,2)-covariant formulation of the boundary behaviour of massless symmetric tensor gauge fields on AdS(d+1) spacetime is obtained. As a byproduct, we identify some useful ambient formulation for Fronsdal fields, conformal currents and shadow fields along with a concise generating-function formulation of the Fradkin-Tseytlin conformal fields somewhat similar to the one obtained by Metsaev. We also show how this approach extends to more general gauge theories and discuss its relation to the unfolded derivation of the boundary dynamics recently proposed by Vasiliev.