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

Symmetries for Galileons and DBI scalars on curved space