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Symmetries for Galileons and DBI scalars on curved space

Garrett Goon, Kurt Hinterbichler, Mark Trodden

TL;DR

The paper develops a brane-based construction that unifies Galileon and DBI scalar theories on curved spaces by embedding a 3-brane in maximally symmetric higher-dimensional bulks. A finite set of second-order, symmetry-protected actions ${\cal L}_1$–${\cal L}_5$ (including a tadpole) arises from Lovelock boundary terms, with the bulk isometries inducing nonlinear shift-like symmetries for the 4D scalar $\pi$. Six maximally symmetric embeddings yield curved-space Galileon and conformal Galileon theories on $AdS_4$, $M_4$, and $dS_4$, each with symmetry-breaking patterns that preserve second-order dynamics and produce curvature-tied potentials. The work shows how symmetry constraints fix masses and enable natural cosmological applications, including inflation and late-time acceleration, while also highlighting possibilities for tachyonic/ghost instabilities and spontaneous symmetry breaking in different backgrounds.

Abstract

We introduce a general class of four-dimensional effective field theories which include curved space Galileons and DBI theories possessing nonlinear shift-like symmetries. These effective theories arise from purely gravitational actions for 3-branes probing higher dimensional spaces. In the simplest case of a Minkowski brane embedded in a higher dimensional Minkowski background, the resulting four-dimensional effective field theory is the Galileon one, with its associated Galilean symmetry and second order equations. However, much more general structures are possible. We construct the general theory and explicitly derive the examples obtained from embedding maximally symmetric branes in maximally symmetric ambient spaces. Among these are Galileons and DBI theories with second order equations that live on de Sitter or anti-de Sitter space, and yet retain the same number of symmetries as their flat space counterparts, symmetries which are highly non-trivial from the 4d point of view. These theories have a rich structure, containing potentials for the scalar fields, with masses protected by the symmetries. These models may prove relevant to the cosmology of both the early and late universe.

Symmetries for Galileons and DBI scalars on curved space

TL;DR

The paper develops a brane-based construction that unifies Galileon and DBI scalar theories on curved spaces by embedding a 3-brane in maximally symmetric higher-dimensional bulks. A finite set of second-order, symmetry-protected actions – (including a tadpole) arises from Lovelock boundary terms, with the bulk isometries inducing nonlinear shift-like symmetries for the 4D scalar . Six maximally symmetric embeddings yield curved-space Galileon and conformal Galileon theories on , , and , each with symmetry-breaking patterns that preserve second-order dynamics and produce curvature-tied potentials. The work shows how symmetry constraints fix masses and enable natural cosmological applications, including inflation and late-time acceleration, while also highlighting possibilities for tachyonic/ghost instabilities and spontaneous symmetry breaking in different backgrounds.

Abstract

We introduce a general class of four-dimensional effective field theories which include curved space Galileons and DBI theories possessing nonlinear shift-like symmetries. These effective theories arise from purely gravitational actions for 3-branes probing higher dimensional spaces. In the simplest case of a Minkowski brane embedded in a higher dimensional Minkowski background, the resulting four-dimensional effective field theory is the Galileon one, with its associated Galilean symmetry and second order equations. However, much more general structures are possible. We construct the general theory and explicitly derive the examples obtained from embedding maximally symmetric branes in maximally symmetric ambient spaces. Among these are Galileons and DBI theories with second order equations that live on de Sitter or anti-de Sitter space, and yet retain the same number of symmetries as their flat space counterparts, symmetries which are highly non-trivial from the 4d point of view. These theories have a rich structure, containing potentials for the scalar fields, with masses protected by the symmetries. These models may prove relevant to the cosmology of both the early and late universe.

Paper Structure

This paper contains 24 sections, 127 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. General brane actions and symmetries
  3. A special case
  4. Maximally symmetric cases
  5. Actions with second order equations of motion
  6. The tadpole term
  7. Explicit expressions for the terms
  8. Maximally Symmetric Examples
  9. A Minkowski brane in a Minkowski bulk: $M_4$ in $M_5$ -- DBI Galileons
  10. Killing vectors and symmetries
  11. A Minkowski brane in an anti-de Sitter bulk: $M_4$ in $AdS_5$ -- Conformal Galileons
  12. Killing vectors and symmetries
  13. A de Sitter brane in a Minkowski bulk: $dS_4$ in $M_5$
  14. Killing vectors and symmetries
  15. A de Sitter brane in a de Sitter bulk: $dS_4$ in $dS_5$
  16. ...and 9 more sections

Figures (3)

  • Figure 1: The field $\pi$ measures the brane position with respect to some chosen foliation.
  • Figure 2: Types of maximally symmetric embedded brane effective field theories, their symmetry breaking patterns, and functions $f(\pi)$. The relationships to the Galileon and DBI theories are also noted.
  • Figure 3: $Z_2$ symmetry breaking for the $dS/AdS$ Galileons.