Galileons as Wess-Zumino Terms

TL;DR

The paper shows that galileons arise as Wess-Zumino terms associated with spontaneously broken spacetime symmetries and that their numbers in a given dimension are governed by relative Lie algebra cohomology. By constructing the galileon and multi-galileon algebras and computing the relevant (d+1)-form co-cycles, it explains why there are a finite set of galileon terms in each dimension and why certain variants (e.g., DBI galileons) are not WZ terms. The conformal case yields a single WZ term (L3) in 4D, while DBI galileons are generally coset-constructible with the tadpole exception. The work provides a unifying algebraic/topological framework linking coset realizations, brane pictures, and the special status of galileons, with potential connections to their non-renormalization properties and to broader topological aspects of symmetry breaking.

Abstract

We show that the galileons can be thought of as Wess-Zumino terms for the spontaneous breaking of space-time symmetries. Wess-Zumino terms are terms which are not captured by the coset construction for phenomenological Lagrangians with broken symmetries. Rather they are, in d space-time dimensions, d-form potentials for (d+1)-forms which are non-trivial co-cycles in Lie algebra cohomology of the full symmetry group relative to the unbroken symmetry group. We introduce the galileon algebras and construct the non-trivial (d+1)-form co-cycles, showing that the presence of galileons and multi-galileons in all dimensions is counted by the dimensions of particular Lie algebra cohomology groups. We also discuss the DBI and conformal galileons from this point of view, showing that they are not Wess-Zumino terms, with one exception in each case.