Complex Function & Spin Connection: What Changes?

[samuelphysics]

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.

 

[haushofer]

You're adding a scalar and a vector now, which i don't know how to interpret. If the metric is real, then the spin connection is also real. So the complex conjugate does nothing. In taking conjugates, you should compare the indices with each other. If you want to take e.g. the Hermitian conjugate on the ij indices of omega, then you should think of what the ij indices of z look like.