# Spin connection property

A simple question: If we have $$z$$ is a complex function, and we have here $$\omega_\mu^{ij}$$ represents some spin connection where $$\mu$$ is spacetime corrdinate.

And say we have $$z + \omega_\mu^{12}$$ no matter for now what the metric is, if I want to take the conjugate of this, is the spin connection affected in any sort of way? So for example I know that$$z ---> \bar{z}$$ but what happens to the spin connection? Does it change sign or does it change indices? I am just wondering.

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