Electric? Then it is geometric

In this work, we revisit Kaluza-Klein theory from the perspective of the classification of elementary particles based on the coadjoint orbit method. We propose a symmetry group for which the electric charge is invariant and, on this basis, a cosmological scenario in which the three former spatial dimensions inflate quickly while the fifth one shrinks, leading to a 4D era where the particles correspond to the coadjoint orbits of this group. By this mechanism, the elementary particles can acquire electric charge as a by-product of the 4 + 1 symmetry breaking of the Universe. By pullback over the space-time, we construct the non-Riemannian connection corresponding to this symmetry group, allowing to recover conservation of the charge and the equation of motion with the Lorentz force. On this ground, we develop a five dimensional extension of the variational relativity allowing to deduce in the classical limit Maxwell's equation.