When dealing with the Dirac equation in curved spacetime one has to replace the partial derivative with the following covariant derivative:
${\partial_{\mu}}-\frac{i}{4}\omega_{\mu}^{\alpha\beta}\sigma_{\alpha\beta}$
where $\omega_{\mu}^{\alpha\beta}$ is the spin connection. What type of physical field this spin connection corresponds to? In the case of the electroweak theory the gauge fields in the covariant derivative correspond to the weak gauge bosons but what actual physical field does the spin connection correspond to?
