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::$omega\_\{mu\}^\{\; ab\}\; =\; e^a\_\; u\; partial\_mu\; e^\{\; u\; b\}\; +\; e^a\_\; u\; e^\{sigma\; b\}\; Gamma^\; u\_\{sigmamu\}$

where $Gamma^\; u\_\{sigmamu\}$ is the Levi-Civita connection and the $e^a\_\; u$ 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.

ee also

* Dirac operator
* Cartan connection applications