Notes on prequantization of moduli of $G$-bundles with connection on Riemann surfaces

Let $𝒳\to S$ be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and $ℱ$ a $G$-bundle over $𝒳$ with connection along the fibres $𝒳\to S$. We construct a line bundle with connection $({ℒ}_{ℱ},{\nabla }_{ℱ})$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $({ℒ}_{ℱ},{\nabla }_{ℱ})$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X\times S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $({ℒ}_{ℱ},{\nabla }_{ℱ})$ provides a prequantization of $S$.