Spin connection

In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the Levi-Civita connection. It can also be regarded as the gauge field generated by local Lorentz transformations.

Definition

$\omega^{\ ab}_{\mu}$ is the spin connection:

$\omega_{\mu}^{\ ab} = e^a_\nu \partial_\mu e^{\nu b} + e^a_\nu e^{\sigma b} \Gamma^\nu_{\sigma\mu}$

where $\Gamma^\nu_{\sigma\mu}$ is the Levi-Civita connection and the $e^a_\nu$ are the local Lorentz frame fields or vierbein. Note that here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices.

See also

• Dirac operator
• Cartan connection applications