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

Marc Henneaux, Cristian Martinez, Ricardo Troncoso, Jorge Zanelli

TL;DR

<3-5 sentence high-level summary>This paper analyzes anti-de Sitter gravity with a minimally coupled self-interacting scalar field in D>=4, focusing on scalar masses in the BF window where slow fall-off modes can back-react on the geometry. It develops relaxed, AdS-invariant boundary conditions that force a functional relation between the scalar expansion coefficients a and b, ensuring finite, integrable Hamiltonian charges that include scalar contributions. The authors derive explicit asymptotic expansions for the scalar and metric fields across multiple mass ranges (including four-dimensional cases with nonlinear back-reaction) and show how logarithmic branches can arise at critical masses, while maintaining overall AdS symmetry and charge conservation. The work extends prior results, demonstrates the compatibility with equations of motion, and discusses higher-dimensional generalizations and potential holographic interpretations involving multi-trace deformations.

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).

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

TL;DR

<3-5 sentence high-level summary>This paper analyzes anti-de Sitter gravity with a minimally coupled self-interacting scalar field in D>=4, focusing on scalar masses in the BF window where slow fall-off modes can back-react on the geometry. It develops relaxed, AdS-invariant boundary conditions that force a functional relation between the scalar expansion coefficients a and b, ensuring finite, integrable Hamiltonian charges that include scalar contributions. The authors derive explicit asymptotic expansions for the scalar and metric fields across multiple mass ranges (including four-dimensional cases with nonlinear back-reaction) and show how logarithmic branches can arise at critical masses, while maintaining overall AdS symmetry and charge conservation. The work extends prior results, demonstrates the compatibility with equations of motion, and discusses higher-dimensional generalizations and potential holographic interpretations involving multi-trace deformations.

Abstract

We examine anti-de Sitter gravity minimally coupled to a self-interacting scalar field in dimensions when the mass of the scalar field is in the range . Here, is the AdS radius, and 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 , 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).

Paper Structure

This paper contains 23 sections, 90 equations.

Table of Contents

  1. Introduction
  2. Standard asymptotic conditions for gravity and charge generators
  3. Gravity and scalar fields saturating the Breitenlohner-Freedman bound
  4. Asymptotically AdS gravity with a minimally coupled scalar field - The case of four dimensions
  5. $m^2_{\ast} + \frac{81}{100 \,l^2} <m^2< m^2_{\ast} +\frac{1}{ l^2}$
  6. $m^2_{\ast} + \frac{9}{16 \, l^2} < m^2 < m^2_{\ast} +\frac{81}{100 \, l^2}$
  7. $m^2_{\ast} + \frac{1}{4 \, l^2} < m^2 < m^2_{\ast} +\frac{9}{16 \, l^2}$
  8. $m^2_{\ast} < m^2 < m^2_{\ast} +\frac{1}{4 \, l^2}$
  9. Asymptotic AdS symmetry
  10. Anti-de Sitter Charges
  11. Compatibility with equations of motion
  12. Higher dimensions
  13. $D=5, 6$, $m^2_{\ast} + \frac{(D-1)^2}{36 \, l^2} < m^2 <m^2_{\ast} + \frac{1}{l^2}$
  14. $D = 5,6$, $m^2_{\ast} < m^2 < m^2_{\ast} + \frac{(D-1)^2}{36 \, l^2}$ and $D \geq 7$
  15. Symmetries and Generators
  16. ...and 8 more sections