Group systems, groupoids, and moduli spaces of parabolic bundles

Abstract

Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification with moduli spaces of holomorphic vector bundles (see [29]). Atiyah and Bott [2] initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over Riemann surfaces, which they analyzed by methods of gauge theory. In particular, they showed that an invariant inner product on the Lie algebra of the Lie group in question induces a natural symplectic structure on a certain smooth open stratum. Although this moduli space is a finite-dimensional object, generally a stratified space which is locally semialgebraic [19] but sometimes a manifold, its symplectic structure (on the stratum just mentioned) was obtained by applying the method of symplectic reduction to the action of an infinite-dimensional group (the group of gauge transformations) on an infinite-dimensional symplectic manifold (the space of all connections on a principal bundle).