Geometrical Heavy Lifting: Yang-Mills, Spin, and Torsion in Dynamical Projective Gravitation

Abstract

Thomas-Whitehead (TW) gravity is a gauge theory of gravitation based on projective geometry. The theory maintains projective symmetry through the TW connection, an affine connection over the volume bundle of the spacetime manifold. TW gravity obtains dynamics through Lovelock expansions in the action while preserving general relativity as a weak field limit. In this paper we clarify the process of lifting tensor and spinor fields from spacetime to the volume bundle and demonstrate that a choice of lifting amounts to a gauge fixing condition. This leads to a natural extension of previous work, where we now realize these prior constructions have been restricted to a particular gauge. In pursuit of generality, we also introduce torsion to the TW connection, leading to new dynamics. In particular, the appearance of torsion induces interaction terms involving gravitational coupling with Yang-Mills fields and Dirac spinors. An explicit realization of this is a geometrically sourced chiral mass term arising from the torsion dynamics.