On the notions of energy tensors in tetrad-affine gravity

We are concerned with the precise modalities by which mathematical constructions related to energy-tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor—the canonical tensor and the stress-energy tensor—exactly coincide with no need for tweaking. Moreover we show how both notions of energy-tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and a spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy-conservation in the presence of torsion.