I'm trying to learn about spin bundles and find myself very confused by the construction and motivation of the spin connection.
The places where I've seen the spin connection formulated, it seems to be explained by the need to account for the change in the vierbeins - i.e. the rotation of the orthonormal frames from point-to-point. But I don't understand why this is related to spinors. It seems to me that an object which carries a frame index just indicates a vector in the tangent space whose components are considered with reference to the orthonormal basis. So a covariant derivative of a vector component should include a factor of the spin connection whenever you are considering such a component. The frame bundle's associated vector bundle under the fundamental representation is just the tangent space. Why does a frame index indicate a half-integer spin?
As I understand it, the spin frame bundle is a lift of the frame bundle using a double cover map. Where does the connection there come from? Is it a pullback of the one on the frame bundle, or do we construct a new one?