I am following [this][1] derivation for the covariant derivative of spinors. I have some questions about this derivation:
1. On page 3 they use the fact, that
\begin{align*}
V^a(x) = \bar{\Psi}(x)\gamma^a\Psi(x)
\end{align*}
transforms as a vector. Its not specified what exactly $\bar{\Psi}(x)$ means. Is it the conjugate transpose of $\Psi$? It definitely cant be the complex conjugate of the spinor since it has to be a row vector.
2. I also dont get how they come up with the equality in Equation (19). When I use the parallel transport equation above for the spinor I come up with
\begin{align*}
S(x+dx)-S(x) = \bar{\Psi}(x)[\Omega_\nu(x)+\bar{\Omega}_\nu(x)]\Psi(x)dx^\nu
\end{align*}
Where do the gamma matrices rise from?


  [1]: https://arxiv.org/abs/1607.04821