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Holonomies of Yang–Mills connections for bundles on a surface with disconnected structure group

Published online by Cambridge University Press:  24 October 2008

Johannes Huebschmann
Johannes Huebschmann
Affiliation:
Université des Sciences et Technologie de Lille, U.F.R. de Mathématiques, F-59 655 Villeneuve d'Ascq, France
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Abstract

Let Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, let

be the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 375 - 384
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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