Pure gauge theory for the gravitational spin connection

TL;DR

This paper explores a novel approach to gravity by treating the spin connection as a dynamical gauge field in a pure Yang-Mills framework, leading to new fermion interactions.

Contribution

It introduces a pure Yang-Mills theory for the gravitational spin connection with dynamical properties, contrasting with traditional metric-based formulations.

Findings

Computed tree-level fermion scattering amplitude via spin connection exchange.

Found a chiral four-Fermi interaction involving right-handed currents.

Identified pathologies and challenges in non-compact gauge theories.

Abstract

The gravitational spin connection appears in gravity as a non-Abelian gauge field for the Lorentz group $SO(3,1)$, which is non-compact. The action for General Relativity is linear in the field strength associated to the spin connection, and its equation of motion corresponds to the standard metricity constraint. Consequently, the zero-torsion spin connection is never realized as an independent degree of freedom and is determined by the vierbein field. In this work, we take a different perspective and consider a pure Yang-Mills theory for the spin connection coupled to Dirac fermions, resulting in the former being a dynamical field. After discussing various approaches towards managing the pathologies associated with non-compact gauge theories, we compute the tree-level amplitude for fermion scattering via a spin connection exchange. In contrast to integrating out torsion in the presence of fermions, the model induces a chiral four-Fermi like term that involves a right-right current interaction, which is not present in the Standard Model.