  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.
