The off-shell behaviour of propagators and the Goldstone field in higher spin gauge theory on $AdS_{d+1}$ space

TL;DR

The paper analyzes the off-shell structure and gauge dependence of bulk-to-bulk propagators for massless higher-spin fields on $AdS_{d+1}$ and clarifies the classical origin of the Goldstone mode. It develops a de Donder gauge-based construction within Fronsdal theory, decomposing the field into a physical traceless part and a Goldstone sector, and studies a one-parameter family of gauges to control trace and transversality. An explicit bitensor expansion for the propagator is derived, with the leading scalar-like function $F_{0}(\zeta)$ solving a wave equation of dimension $\Delta_{\ell}=\ell+d-2$, and the bulk-to-boundary limit reduces to Dobrev’s propagator without trace terms, yielding the correct boundary CFT data after projection. The work thus connects bulk higher-spin dynamics to boundary conserved currents, clarifies mass-generation via the Goldstone mode, and reinforces the consistency of $AdS/CFT$ for higher-spin theories at the level of propagators.

Abstract

A detailed analysis of the structure and gauge dependence of the bulk-to-bulk propagators for the higher spin gauge fields in $AdS$ space is performed. The possible freedom in the construction of the propagators is investigated and fixed by the correct boundary behaviour and correspondence to the representation theory results for the $AdS$ space isometry group. The classical origin of the Goldstone mode and its connection with the gauge fixing procedure is considered.