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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?

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