I am reading a paper written by physicists and they say the following:
Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If $e^{f_{ab}}$ are the transition functions of $L$ on the non-empty overlapping of two open sets $U_{a}$ and $U_{b}$, then the transition functions in the associated $U(1)$-bundle $\mathcal{U}$ are $e^{iIm(f_{ab})}$.
In addition, let $\theta$ be the local one-form connection of $\nabla$ on a local trivialization of $L$. Then corresponding connection on the $U(1)$-bundle $\mathcal{U}$ is $Im(\theta) = -\frac{i}{2}(\theta - \bar{\theta})$. Then, the covariant derivative of any section $\Psi\in\Gamma(\mathcal{U})$ is $d\Psi + Im(\theta)\Psi$.
My question is: is this correct? Because I cannot make sense of it.