Asymptotic behavior and Hamiltonian analysis of anti-de Sitter gravity coupled to scalar fields

TL;DR

This paper investigates the asymptotic behavior of anti-de Sitter gravity coupled to scalar fields, establishing invariant conditions for well-defined Hamiltonian generators despite scalar back reaction and logarithmic branches.

Contribution

It introduces new asymptotic conditions for AdS gravity with scalar fields that ensure finite Hamiltonian generators and analyzes the effects of scalar self-interactions and back reaction.

Findings

Asymptotic conditions preserving AdS invariance are formulated.

Scalar field contributions modify the standard asymptotic charges.

Logarithmic branches occur for specific scalar masses, affecting the asymptotic structure.

Abstract

We examine anti-de Sitter gravity minimally coupled to a self-interacting scalar field in $D\geq 4$ dimensions when the mass of the scalar field is in the range $m_{\ast}^{2}\leq m^{2}<m_{\ast} ^{2}+l^{-2}$. Here, $l$ is the AdS radius, and $m_{\ast}^{2}$ is the Breitenlohner-Freedman mass. We show that even though the scalar field generically has a slow fall-off at infinity which back reacts on the metric so as to modify its standard asymptotic behavior, one can still formulate asymptotic conditions (i) that are anti-de Sitter invariant; and (ii) that allows the construction of well-defined and finite Hamiltonian generators for all elements of the anti-de Sitter algebra. This requires imposing a functional relationship on the coefficients $a$, $b$ that control the two independent terms in the asymptotic expansion of the scalar field. The anti-de Sitter charges are found to involve a scalar field contribution. Subtleties associated with the self-interactions of the scalar field as well as its gravitational back reaction, not discussed in previous treatments, are explicitly analyzed. In particular, it is shown that the fields develop extra logarithmic branches for specific values of the scalar field mass (in addition to the known logarithmic branch at the B-F bound).