Boundary action of free AdS higher-spin gauge fields and the holographic correspondence

TL;DR

This work determines the boundary terms needed to reproduce the AdS Fronsdal equations for free higher-spin fields with a finite boundary and analyzes the gauge-invariance constraints that arise. It shows that, for spins $s>2$, no local boundary counterterm fully preserves bulk gauge symmetry, with the residual HS Weyl transformations corresponding to boundary conformal HS symmetries and giving rise to a HS Weyl anomaly in even boundary dimensions. The authors express the on-shell AdS higher-spin action as a functional of a boundary conformal higher-spin field $h^{(s)}$, obtaining explicit finite or logarithmic contributions that match the quadratic part of the boundary CFT effective action, up to local counterterms. This work strengthens the HS/CFT correspondence at the quadratic level and highlights how boundary terms encode Weyl anomalies, suggesting rich structures for higher-spin interactions and non-Abelian deformations in the bulk.

Abstract

We determine the boundary terms of the free higher-spin action which reproduce the AdS Fronsdal equations in an AdS manifold with a finite distance boundary. The boundary terms are further constrained by the gauge invariance of the total action. We show that, for spins larger than two, no local boundary term can restore the full gauge symmetry, and the broken symmetry corresponds to higher-spin Weyl transformations on the boundary CFT. The boundary action is used for the evaluation of the on-shell higher-spin AdS action in terms of the boundary data given by a conformal higher-spin field.