Spin connection

Summary

The spin connection of a pseudo-Riemannian manifold is the Levi-Civita connection written in a local orthonormal frame. It is a matrix of 1-forms with values in the Lie algebra of the orthogonal group of the metric. Because it takes values in this Lie algebra, it lifts to the spin group. So it defines a covariant derivative of spinor fields, which is where its name comes from. It is the basic tool to couple fermions to gravity and to write the Dirac operator on a curved spinor bundle.