Explicit Determination of the Picard Group of Moduli Spaces of Semi-Stable G-Bundles on Curves

Let $\mathcal{C}$ be a smooth irreducible projective curve over the complex numbers and let $G$ be a simple simply-connected complex algebraic group. Let $\mathfrak{M}=\mathfrak{M}(G,\mathcal{C})$ be the moduli space of semistable principal $G$-bundles on $\mathcal{C}$. By an earlier result of Kumar-Narasimhan, the Picard group of $\mathfrak{M}$ is isomorphic with the group of integers. However, in their work the generator of the Picard group was not determined explicitly. The aim of this paper to give the generator `explicitly.’ The proof involves an interesting mix of geometry and topology.