The spin connection $\omega^a_{b\nu}$ is used to define the covariant derivative of a spinor in curved spacetime. I want to explicitly calculate the covariant derivative:
$$\nabla_\nu\Psi=(\partial_\nu+\Omega_\nu)\Psi$$
for a given metric in two dimensions. In this article here, they derive $\Omega_\nu$ to be:
$$\Omega_\nu = \frac{1}{8}\omega_{ab\nu}[\gamma^a,\gamma^b] \ ,$$
where $\gamma^a$ are the gamma matrices in flat spacetime. They also show, that the spin connection coefficients can be obtained using the Christoffel symbols:
\begin{align*} \omega^a_{b\nu}=e^a_\mu\partial_\nu(e^\mu_b)+e^a_\mu e^\sigma_b\Gamma^\mu_{\sigma\nu} \end{align*}
I can show, that for the metric $\omega^1_{00}=\omega^0_{10} = \tan(\theta)$ and all other coefficients vanish. If I try to explicitly compute $\Omega_\nu$, I just get $\Omega_0 = \Omega_1 = 0$, which doesn't make sense in a curved spacetime. My question here is, how are $\omega^a_{b\nu}$ and $\omega_{ab\nu}$ are related, since I guess this is my mistake here. Just like the Christoffel symbols, the spin connection is not a tensor. So what does it mean to raise or lower its indices?
