 Spin connection

In an orthonormal frame, the spin connection is the Lorentz connection acting on spinors. With ${\gamma }^{AB}=\frac{1}{2}[{\gamma }^A,{\gamma }^B]$, its covariant derivative is ${\nabla }_{\mu }\psi ={\partial }_{\mu }\psi +\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}\psi$. The metric connection determines $\omega$ when torsion conditions are specified. A zweibein gives its two-dimensional form; omitting it on a curved worldsheet generally destroys covariance.

The spin connection is a mathematical construct used in the context of differential geometry and gauge theories, particularly in the study of general relativity and theories involving spinors, such as quantum field theories in curved spacetime. It is essential for describing how spinor fields (which are fields that transform under the spin group and are important in particle physics) behave in curved spacetime.