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

We consider a self-interacting scalar field whose mass saturates the BreitenlohnerFreedman bound, minimally coupled to Einstein gravity with a negative cosmological constant in D ≥ 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 backreaction 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. ∗ E-mail: [email protected], [email protected], [email protected], [email protected]