In this question- Connection of $\mathcal{O}(n)$ on a toric manifold, it is explained that the covariant derivative on $\mathcal{O}(n)$ is given by $$ \nabla=d+nA, $$ where $A$ is the connection on a $U(1)$ bundle, $P$, such that the covariant derivative on $\mathcal{O}(1)$ is $\nabla=d+A$.
How does this generalize to $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$, which is discussed here - Proof of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is ample $\iff$ $a,b >0$. ?
Would it be correct to say that the line bundle $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is associated to a $U(1)\times U(1)$ bundle, whereby the covariant derivative is given by $$ \nabla=d+nA^{(1)}+mA^{(2)}, $$ where $A^{(1)}$ and $A^{(2)}$ are the components of the $U(1)\times U(1)$ connection, such that the covariant derivative on $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(1,1)$ is $\nabla=d+A^{(1)}+A^{(2)}$?
References would be appreciated.