Extended Gauss-Bonnet gravities in Weyl geometry

In this paper we consider an extended Gauss-Bonnet gravity theory in arbitrary dimensions and in a space provided with a Weyl connection, which is torsionless but not metric-compatible, the non-metricity tensor being determined by a vector field. The considered action consists of the usual Einstein-Hilbert action plus all the terms quadratic in the curvature that reduce to the usual Gauss-Bonnet term for vanishing Weyl connection, i.e., when only the Levi-Civita part of the connection is present. We expand the action in terms of Riemannian quantities and obtain vector-tensor theories. We find that all the free parameters only appear in the kinetic term of the vector field so that two branches are possible: one with a propagating vector field and another one where the vector field does not propagate. We focus on the propagating case. We find that in 4 dimensions, the theory is equivalent to Einstein's gravity plus a Proca field. This field is naturally decoupled from matter so that it represents a natural candidate for dark matter. Also in d=4, we discuss a non-trivial cubic term in the curvature that can be constructed without spoiling the second order nature of the field equations because it leads to the vector-tensor Horndeski interaction. In arbitrary dimensions, the theory becomes more involved. We show that, even though the vector field presents kinetic interactions which do not have U(1) symmetry, there are no additional propagating degrees of freedom with respect to the usual massive case. Interestingly, we show that this relies on the fact that the corresponding Stueckelberg field belongs to a specific class within the general Horndeski theories. Finally, since Weyl geometries are the natural ground to build scale invariant theories, we apply the usual Weyl-gauging in order to make the Horndeski action locally scale invariant and discuss on new terms that can be added.