# How does the spin connection affect the dynamics of a fermion in curved space?

Consider a massless right-handed Majorana fermion in curved spacetime. Without any other fields present, the Lagrangian density is (I believe) the following:

$$\mathcal{L}_{\psi} = \sqrt{g}i\bar{\psi}_R\gamma^\mu\nabla_\mu\psi_R \quad \text{where} \quad \nabla_\mu\psi_R = \left(\partial_\mu + \frac{i}{2}\chi^{ab}_\mu\Sigma_{ab}\right)\psi_R .$$

$$\chi^{ab}_\mu$$ are the spin connection coefficients and $$\Sigma_{ab} = \frac{i}{4}[\gamma_a,\gamma_b]$$ where $$\gamma_c$$ are the Lorentz generators.

My question is, how does this spin connection affect the fermion dynamics? Can we think of $$\chi_\mu(x) \equiv \chi^{ab}_\mu(x)\Sigma_{ab}$$ as a regular gauge field (albeit with a geometric origin)? How does this spin connection affect loop calculations, etc.?