Affine Gauge Theory: A Diffeomorphism Invariant Gauge Theory of Gravity

TL;DR

Affine Gauge Theory (AGT) reframes gravity as a gauge theory with the affine group on an Affine Bundle, producing a local curvature $\tilde{F}$ that decomposes as $\tilde{F}=F+T$, where $F$ is Frame Bundle curvature and $T$ is torsion; in the TEGR regime, $F=0$ giving $\tilde{F}=T$, a genuine gauge-theoretic realization of TEGR. The work establishes a rigorous Affine–Frame Bundle correspondence via maps $\beta$ and $\gamma$, and proves both Translational Gauge Invariance and Diffeomorphism Invariance of the essential structures (canonical 1-form and flat Frame Bundle connection), extending these invariances to TEGR and AGT as theories. It further shows how diffeomorphism invariance arises on the Affine Bundle through the soldering map $\gamma$, and clarifies the relation between Translational Gauge Invariance and Diffeomorphism Invariance in the gravity context. Collectively, the framework offers a background-independent, gauge-theoretic description of gravity with clear symmetry structure and potential implications for unification and quantum gravity.

Abstract

This paper is a comprehensive investigation of the Affine Gauge Theory (AGT) as a gauge theory of gravity having the same mathematical structure as gauge theories of the other fundamental forces of nature. This mathematical structure consists of a principal fiber bundle over the spacetime manifold that is endowed with an affine connection. The relationship between AGT and various formulations of teleparallel theories of gravity, which are alternatives to General Relativity, are examined. Here, it is argued that the Affine Bundle is the most natural principal fibre bundle for a gauge theory of gravity. AGT is also shown to be strictly diffeomorphism invariant. In particular, an explicit proof of diffeomorphism invariance in AGT is given - showing that AGT possesses the important symmetry of General Relativity, as would be expected from a theory of gravity. Lastly, the claim that AGT is background independent, as General Relativity is, from varying degrees of strictness in the definition of background independence is closely examined. Reasons for why a background independent theory is preferred are also discussed.