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

摘要

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