Notes on prequantization of moduli of $G$-bundles with connection on Riemann surfaces
Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 181-186.

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 $\left({ℒ}_{ℱ},{\nabla }_{ℱ}\right)$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $\left({ℒ}_{ℱ},{\nabla }_{ℱ}\right)$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X×S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $\left({ℒ}_{ℱ},{\nabla }_{ℱ}\right)$ provides a prequantization of $S$.

@article{AMBP_2004__11_2_181_0,
author = {Rodriguez, Andres},
title = {Notes on prequantization of moduli of $G$-bundles with connection on Riemann surfaces},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {181--186},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {11},
number = {2},
year = {2004},
doi = {10.5802/ambp.191},
mrnumber = {2109606},
zbl = {1078.53095},
language = {en},
url = {www.numdam.org/item/AMBP_2004__11_2_181_0/}
}
Rodriguez, Andres. Notes on prequantization of moduli of $G$-bundles with connection on Riemann surfaces. Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 181-186. doi : 10.5802/ambp.191. http://www.numdam.org/item/AMBP_2004__11_2_181_0/

[1] Deligne, P. Théorie de Hodge. III., Inst. Hautes Ètudes Sci. Publ. Math., Volume 44 (1974), pp. 5-77 | Article | Numdam | MR 498552 | Zbl 0237.14003