Gravity from Invariant Weyl-Integrable space-time (IWIST)
In this paper we propose a geometrically motivated formulation of gravity in an invariant Weyl--Integrable space-time (IWIST), in which the background geometry is not imposed a priori but is dynamically determined through the Palatini variational principle. For a scalar field non-minimally coupled to gravity action, the resulting compatibility condition defines a Weyl--Integrable geometry which is preserved by a local Weyl transformation of the metric and the geo\-me\-tri\-cal scalar field. We construct an action invariant under both diffeomorphisms and Weyl transformations and introduce a Weyl-covariant variational procedure based on an invariant extension of the divergence theorem. The corresponding field equations are obtained for the metric, the Weyl scalar, and the Weyl gauge field. We further formulate the theory in the Einstein--Riemann frame, where the effective metric is Riemannian and the scalar field is re\-inter\-pre\-ted as a phy\-si\-cal degree of freedom of geometric origin. Finally, we propose a geometrical mechanism for the explicit breaking of Weyl symmetry by introducing a coupling between the Weyl gauge field and a current constructed from the scalar sector. The resulting theory remains diffeomorphism covariant while departing from local Weyl invariance. The Einstein--Riemann representation of the broken theory contains an additional generally covariant interaction, making the Weyl-invariant model a particular limit of a more general gravitational theory.