Fundamental groups of moduli of principal bundles on curves

Abstract

Let X be a compact connected Riemann surface of genus g, with $g\, \ge \, 2$, and let ${\text {G}}$ be a connected semisimple affine algebraic group defined over ${\mathbb {C}}$. Given any $\delta \, \in \, \pi _1({\text {G}})$, we prove that the moduli space of semistable principal ${\text {G}}$-bundles over X of topological type $\delta$ is simply connected. More generally, if ${\text {G}}$ is a connected reductive complex affine algebraic group, then the fundamental group of the moduli space is isomorphic to ${\mathbb Z}^{2gd}$, where d is the complex dimension of the center of ${\text {G}}$. In contrast, the fundamental group of the moduli stack of principal ${\text {G}}$-bundles over X of topological type $\delta$ is shown to be isomorphic to $H^1(X,\, \pi _1({\text {G}}))$, when ${\text {G}}$ is semisimple. We also compute the fundamental group of the moduli stack of principal ${\text {G}}$-bundles when ${\text {G}}$ is reductive.