Scalar potentials, propagators and global symmetries in AdS/CFT

We study the transition of a scalar field in a fixed $AdS_{d+1}$ background between an extremum and a minimum of a potential. We first prove that two conditions must be met for the solution to exist. First, the potential involved cannot be generic, i.e. a fine-tuning of their parameters is mandatory. Second, at least in some region its second derivative must have a negative upper limit which depends only on the dimensionality $d$. We then calculate the boundary propagator for small momenta in two different ways: first in a WKB approximation, and second with the usual matching method, generalizing the known calculation to arbitrary order. Finally, we study a system with spontaneously broken non-Abelian global symmetry, and show in the holographic language why the Goldstone modes appear.
Comment: 26 pages - Invited contribution for the Central European Journal of Physics, topical issue devoted to "Cosmology and Particle Physics beyond Standard Models". Some parts overlap with 1304.3051v1, which has been replaced by the published version