Projective symmetries and induced electromagnetism in metric-affine gravity

We present a framework in which the projective symmetry of the Einstein–Hilbert action in metric-affine gravity is used to induce an effective coupling between the Dirac lagrangian and the Maxwell field. The effective U(1) gauge potential arises as the trace of the non-metricity tensor Qμaa and couples in the appropriate way to the Dirac fields to in order to allow for local phase shifts. On shell, the obtained theory is equivalent to Einstein–Cartan–Maxwell theory in presence of Dirac spinors.