Abstract
The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian p-forms. In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixed-symmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to p-forms and scalars with a shift-symmetry. The simplest case corresponds to linearised gravity with Lovelock invariants, relating the Galileon symmetry to diffeomorphisms. Finally, we examine the coupling of a mixed-symmetry tensor to gravity, and demonstrate in an explicit example that the inclusion of appropriate counterterms retains second order field equations.
Highlights
These include scalar theories with field equations containing derivatives up to second order instead of strictly second order [12]
The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian p-forms
In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixedsymmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to p-forms and scalars with a shift-symmetry
Summary
Given that the actions for Galileons are highly non-linear, they appear rather complicated and employ a large number of indices. Extending the bosonic (degree-0) coordinates (xi) by these degree-1 variables essentially means that we are going to write down actions on a graded (super)manifold M. These variables satisfy the following properties: θiθj = −θjθi , χiχj = −χjχi , θiχj = χjθi. In curved space and for more general choices of coordinates, the epsilon symbol is to be replaced by the epsilon tensor, i.e. the value of integrals like (2.9) can depend on spacetime position. These integral formulas are pivotal in our formulation, since the Lagrangians will always involve integration over the graded variables
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