# Spin connection raise and lower flat indices

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?

• for the metric For what metric? What is $\theta$? Commented Sep 27, 2022 at 16:12
• Qmechanic provided a hint by editing the title to refer to “flat indices”. Commented Sep 27, 2022 at 19:48

## 1 Answer

Flat space indices are raised/lowered with the tangent space metric $$\eta$$. The connection isn't a tensor but we use the notation as shorthand, e.g.

$$\omega_{\mu a}{}^b = \eta_{ac} \omega_{\mu}{}^{cb}$$