In Einstein's Theory, A Rigorous Introduction for the Mathematically Untrained:
The change of the covariant components a vector by parallel transport around an indefinitely small closed curve enclosing a surface with area
$$dS^{\alpha\beta}$$ may now be written
(Equation 9.30) $${\Delta}A_{\mu}=\frac{1}{2}R^{\nu}_{\mu\alpha\beta}A_{\nu}\Delta{S}^{\alpha\beta}$$
Here,
$$R^{\nu}_{\mu\alpha\beta}$$
is the Riemann Curvature Tensor, defined as:
(Equation 9.29) $$R^{\nu}_{\mu\alpha\beta}=\Gamma^{\tau}_{\mu\beta}\Gamma^{\nu}_{\tau\alpha}-\Gamma^{\tau}_{\mu\alpha}\Gamma^{\nu}_{\tau\beta}+\Gamma^{\nu}_{\mu\beta,\alpha}-\Gamma^{\nu}_{\mu\alpha,\beta}$$
But Equation 9.28 says
$${\Delta}A_{\mu}=\left(\Gamma^{\tau}_{\mu\beta}\Gamma^{\nu}_{\tau\alpha}-\Gamma^{\tau}_{\mu\alpha}\Gamma^{\nu}_{\tau\beta}+\Gamma^{\nu}_{\mu\beta,\alpha}-\Gamma^{\nu}_{\mu\alpha,\beta}\right)A_{\nu}\Delta{S}^{\alpha\beta}$$
So where did the factor of one half come from?