Asymptotically Anti-de Sitter spacetimes and scalar fields with a logarithmic branch

TL;DR

The paper analyzes asymptotically anti-de Sitter spacetimes sourced by a self-interacting scalar field whose mass saturates the Breitenlohner-Freedman bound $m_*^2 = -(D-1)^2/(4 l^2)$. It demonstrates that the scalar logarithmic branch back-reacts on the metric, requiring relaxed boundary conditions that still preserve the original symmetry group $SO(D-1,2)$ (and the 2D conformal group in $D=3$) and yield finite, scalar-augmented charges. By deriving explicit relaxed asymptotics and computing the conserved charge $Q(\xi)$ via Regge-Teitelboim, the authors show divergences cancel and the total charge remains finite with a genuine scalar contribution. The work connects to holographic renormalization in AdS/CFT and has implications for energy positivity, supersymmetry compatibility, and cosmic censorship in AdS spacetimes with scalar hair.

Abstract

We consider a self-interacting scalar field whose mass saturates the Breitenlohner-Freedman bound, minimally coupled to Einstein gravity with a negative cosmological constant in D \geq 3 dimensions. It is shown that the asymptotic behavior of the metric has a slower fall-off than that of pure gravity with a localized distribution of matter, due to the back-reaction of the scalar field, which has a logarithmic branch decreasing as r^{-(D-1)/2} ln r for large radius r. We find the asymptotic conditions on the fields which are invariant under the same symmetry group as pure gravity with negative cosmological constant (conformal group in D-1 dimensions). The generators of the asymptotic symmetries are finite even when the logarithmic branch is considered but acquire, however, a contribution from the scalar field.