Fundamental group of moduli of principal bundles on curves

TL;DR

This paper proves that the moduli space of semistable principal G-bundles over a genus ≥2 Riemann surface is simply connected, while the moduli stack's fundamental group relates to the first cohomology of the surface with coefficients in π₁(G).

Contribution

It establishes the fundamental group structure of moduli spaces and stacks of principal G-bundles over curves, revealing their topological properties.

Findings

Moduli space of semistable principal G-bundles is simply connected.

Fundamental group of the moduli stack is isomorphic to H^1(X, π_1(G)).

Results depend on the topological type δ in π_1(G).

Abstract

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $\delta \in \pi_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $\delta$ is simply connected. In contrast, the fundamental group of the moduli stack of principal ${G}$--bundles over $X$ of topological type $\delta$ is shown to be isomorphic to $H^1(X, \pi_1({G}))$.