Conformal defects and Goldstone bosons in Anti-de Sitter space

We study local quantum field theories in Anti-de Sitter (AdS) space, with boundary conditions that break some of the bulk isometries. Specifically, we focus on conformal defects and we prove that their spectrum supports a displacement operator of protected dimension, despite the non-local nature of the conformal theory living at the boundary of AdS. If the defect breaks a global symmetry, a tilt operator is also present. The existence of a displacement was conjectured in arXiv:2508.08250 for Wilson loops in Yang-Mills theories in AdS. Our proof is valid in general and applies, in particular, to defects in long-range models, as we discuss in various examples. In the bulk, the modes sourced by the protected operators have Compton wavelength of order of the AdS radius: they constitute the AdS analogue of the Goldstone bosons for the spontaneous breaking of the corresponding symmetries.