Conservation laws in metric-affine gravitation theories: superpotentials

By applying the machinery of the Lagrangian formalism in field theory, we study the conservation laws of a metric-affine gravitation theory in which the dynamical fields are the spin structures, the linear connections and the fermion fields on a 4-dimensional manifold. The system is assumed to be symmetric with respect to the group of all transformations of the spin bundle. The results obtained are the following. The currents associated with the infinitesimal vertical (internal) transformations of the symmetry group vanish identically. As a consequence, to every vector field on the world manifold there corresponds a well-defined current, namely the energy-momentum current. Moreover, the superpotential term contained in this current is independent of the presence of fermion fields. Actually, the expression we get for the superpotential coincides with that found in the purely metric-affine context and generalizes the well-known expression obtained by Komar.