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.?


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