Geometry of Special Galileon

TL;DR

This paper explores the geometric and symmetry structures of the Special Galileon theory, revealing its hidden symmetries, invariant actions, and geometric interpretation as a brane in higher-dimensional space, advancing understanding of its unique properties.

Contribution

It provides a geometric construction of the Special Galileon, interprets its hidden symmetry as a coset space transformation, and classifies invariant Lagrangians using a brane in higher-dimensional space.

Findings

Identifies a three-parametric family of invariant Galileon actions.

Shows the symmetry as a transformation of the coset space $GAL(D,1)/SO(1,D-1)$.

Relates the symmetry to a $U(1,D-1)$ symmetry of a target space.

Abstract

Theory known as Special Galileon has recently attracted considerable interest due to its peculiar properties. It has been shown that it represents an extremal member of the set of effective field theories with enhanced soft limit. This property makes its tree-level S-matrix fully on-shell reconstructible and representable by means of the Cachazo-He-Yuan representation. The enhanced soft limit is a consequence of new hidden symmetry of the Special Galileon action, however, until now, the origin of this peculiar symmetry has remained unclear. In this paper we interpret this symmetry as a special transformation of the coset space $GAL(D,1)/SO(1,D-1)$ and show, that there exists a three-parametric family of invariant Galileon actions. The latter family is closed under duality which appears as a natural generalization of the above mentioned symmetry. We also present a geometric construction of the Special Galileon action using $D$-dimensional brane propagating in $2D$-dimensional flat pseudo-riemannian space. Within such framework, the Special Galileon symmetry emerges as an $U(1,D-1)$ symmetry of the target space, which can be treated as a $D$-dimensional K\"ahler manifold. Such a treatment allows for classification of the higher order invariant Lagrangians needed as counterterms on the quantum level. We also briefly comment on relation between such higher order Lagrangians and the Lagrangians invariant with respect to the polynomial shift symmetry.