Dynamical dark energy and gravitational coupling from moving geometries

We introduce the notion of moving Cartan geometries described by quotients of Lie groups and Lie algebras with spacetime dependant structure constants and construct associated deformed topological gauge action functionals for Lorentzian (including dS and AdS) and Lorentz$\times$Weyl moving geometries. The actions feature a generalization of the Nieh-Yan topological term for a varying coupling constant. We compute the equations of motion of the gauge $+$ matter actions and show that they dictate at each spacetime point the geometry, leading to both a dynamical source of dark energy and a dynamical gravitational coupling, described by combinations of scalars built from spacetime curvature, torsion and the matter content of the theory. The action becomes asymptotically topological when the gauge action contribution to dark energy vanishes.