Group systems, groupoids, and moduli spaces of parabolic bundles

TL;DR

This paper extends a symplectic construction to punctured surfaces, producing stratified symplectic moduli spaces of flat G-bundles with conjugacy class conditions, related to parabolic bundle moduli spaces.

Contribution

It generalizes a finite-dimensional symplectic construction to punctured surfaces, linking flat G-bundles, moduli spaces, and parabolic bundles with new stratified symplectic structures.

Findings

Constructs a symplectic structure on moduli spaces with punctures.

Establishes a Hamiltonian G-action with a reduced moduli space.

Relates the moduli spaces to semistable parabolic bundles.

Abstract

Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $\pi$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of conjugacy classes in $G$, one for each puncture. A finite-dimensional construction used earlier to obtain a symplectic structure on the moduli space of flat $G$-bundles over compact $M$ is extended to the punctured case. It yields a symplectic structure on a certain smooth manifold $\Cal M_{\bold C}$ containing the space $\roman{Hom}(\pi,G)_{\bold C}$ of homomorphisms mapping the generators corresponding to the punctures into the corresponding conjugacy classes. It also yields a Hamiltonian $G$-action on $\Cal M_{\bold C}$ such that the reduced space equals the moduli space $\roman{Rep}(\pi,G)_{\bold C}$ of representations. For $G$ compact, each such space, obtained by finite-dimensional symplectic reduction, is a {\it stratified symplectic space\/}. For $G=U(n)$ one gets moduli spaces of semistable holomorphic parabolic bundles or spaces closely related to them.