Dirac Spinors on Generalised Frame Bundles: a frame bundle formulation for Einstein-Cartan-Dirac theory

TL;DR

This paper develops a geometric formulation of Einstein-Cartan-Dirac theory using generalized frame bundles, clarifying the structure and extending the model to include Dirac spinor fields coupled to torsion.

Contribution

It provides a new geometric framework for Einstein-Cartan-Dirac theory based on generalized frame bundles, extending previous formulations to include spinor fields.

Findings

Established a local equivalence to a frame bundle structure.

Derived variational principles inducing a bundle structure over a base manifold.

Extended the Einstein-Cartan model to incorporate Dirac spinor fields.

Abstract

We clarify the structure obtained in H\'elein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle starting from a structure-less differentiable 10-manifold (arXiv:1508.07765v2 [math-ph]). The obtained structure is locally equivalent to a frame bundle. We show that under global hypotheses the formalism allows to produce variational principles which induce a frame bundle structure with a fibration over a base manifold and a variational principle on the base manifold. Our perspective on the model allows us to extend the Einstein-Cartan model to the Einstein-Cartan-Dirac theory which includes a Dirac spinor field coupled to the torsion field.