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Group systems, groupoids, and moduli spaces of parabolic bundles.(English)Zbl 0885.58011

The authors describe the moduli space of homomorphisms from systems of fundamental groups of (punctured) surfaces to compact connected Lie groups, which coincides with the moduli space of parabolic holomorphic vector bundles. In particular, they construct a symplectic structure on the top stratum of the moduli space using purely finite dimensional methods.M. F. Atiyah andR. Bott initiated an infinite dimensional approach using gauge theory and holomorphic vector bundles in Philos. Trans. R. Soc. Lond., Ser. A 308, 523-615 (1983;Zbl 0509.14014).
The main results of the paper are the following: The moduli space can be obtained by symplectic reduction from a finite dimensional symplectic manifold. The symplectic form is described in terms of the cup product on group cohomology with values in the Lie algebra. The symplectic structure coincides with the one obtained via gauge theory. The finite dimensional approach implies that some of the results are also valid for noncompact Lie groups. The principal idea is the use of group systems and fundamental groupoids instead of the fundamental group, to allow the boundary of the surface to be non void.

MSC:

58D27 Moduli problems for differential geometric structures
58B25 Group structures and generalizations on infinite-dimensional manifolds
58D15 Manifolds of mappings
32G13 Complex-analytic moduli problems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0509.14014

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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