Fundamental group of moduli of principal bundles on curves
Indranil Biswas, Swarnava Mukhopadhyay, Arjun Paul

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 be a compact connected Riemann surface of genus at least two, and let be a connected semisimple affine algebraic group defined over . For any , we prove that the moduli space of semistable principal --bundles over of topological type is simply connected. In contrast, the fundamental group of the moduli stack of principal --bundles over of topological type is shown to be isomorphic to .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Geometry and complex manifolds
