Multi-galileons, solitons and Derrick's theorem

TL;DR

The paper extends Galilean-invariant scalar theories to multi-component fields with internal $SO(N)$ and $SU(N)$ symmetries, constructing a finite and manageable set of multi-galileon terms that preserve second-order equations of motion. For each representation (SO(N) fundamental/adjoint and SU(N) fundamental/adjoint), it identifies the complete, symmetry-allowed terms in four dimensions and provides explicit Lagrangians, along with degeneracy relations for small N. By revisiting Derrick's theorem, it demonstrates how higher-gradient galileon interactions can support static solitons when fields are constrained on nonlinear manifolds, and presents a concrete $SO(4)$ hedgehog solution on $S^3$ with finite energy, analyzed for stability. Overall, the work shows that symmetry constraints yield a minimal, tractable set of multi-galileon terms and opens pathways to stable solitons within these generalized frameworks, with future work on stability analysis and extensions to other dimensions.

Abstract

The field theory Galilean symmetry, which was introduced in the context of modified gravity, gives a neat way to construct Lorentz-covariant theories of a scalar field, such that the equations of motion contain at most second-order derivatives. Here we extend the analysis to an arbitrary number of scalars, and examine the restrictions imposed by an internal symmetry, focussing in particular on SU(N) and SO(N). This therefore extends the possible gradient terms that may be used to stabilise topological objects such as sigma model lumps.

Figures (1)

• Figure 1: Static solitonic solutions for winding number $n=1$ (solid-blue line) and $n=2$ (dashed-red line). The spherically symmetic ansatz (\ref{['ansatz']}) is used. The energy of them are $E^{(1)}\simeq 101.8$ and $E^{(2)}\simeq 462.3$ respectively.