# Kinetic term for a Majorana fermion in curved Weyl geometry

I am trying to write the action for a Majorana fermion on a curved Weyl-gravity background. Since I am considering a fermion in curved space, the tetrad formalism is appropriate and the kinetic term for a right-handed fermion should be the following. $$\mathcal{L} = \sqrt{g}i\bar{\psi}_R\gamma^a e^\mu_a \nabla_\mu\psi_R$$ My question is, how does the covariant derivative look when we are considering (non-Riemannian) Weyl gravity?

By Weyl gravity, I mean the formulation where we pick up a gauge field $$\omega_\mu$$ of geometric origin when we enforce scale invariance as a local symmetry (see for example https://arxiv.org/abs/1812.08613).

It seems to me that there are two separate connections contributing to the derivative here:

1. Spin connection $$\chi^{ab}_\mu$$: $$\nabla^{(1)}_\mu \psi_R = (\partial_\mu + \chi^{ab}_\mu\Sigma_{ab})\psi_R$$
2. Weyl gauge connection $$\omega_\mu$$: $$\nabla^{(2)}_\mu \psi_R = (\partial_\mu - \frac{\lambda}{2}\omega_\mu)\psi_R$$

How exactly do these two contributions combine? Is it as simple as the sum $$\nabla_\mu \psi_R= (\partial_\mu + \chi^{ab}_\mu\Sigma_{ab} - \frac{\lambda}{2}\omega_\mu)\psi_R$$, or is there an interaction between the spin connection and the gauge field of some kind?