Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-08T22:06:39.857Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Atiyah, M. and Bott, R.. The Yang–Mills equations over Riemann surfaces. Phil. Trans. R. Soc. London A 308 (1982), 523–615.Google Scholar
[2]Huebschmann, J.. The singularities of Yang–Mills connections for bundles on a surface. I. The local model (preprint).Google Scholar
[3]Huebschmann, J.. The singularities of Yang–Mills connections for bundles on a surface. II. The stratification (preprint).Google Scholar
[4]Huebschmann, J.. The singularities of Yang–Mills connections for bundles on a surface. III. The identification of the strata (in preparation).Google Scholar
[5]Huebschmann, J.. Smooth structures on moduli spaces of central Yang–Mills connections for bundles on a surface (preprint).Google Scholar
[6]Huebschmann, J.. Poisson structures on certain moduli spaces for bundles on a surface (preprint).Google Scholar
[7]Huebschmann, J.. Poisson geometry for flat connections for SJJ (2)-bundles on surfaces (preprint).Google Scholar
[8]Huebschmann, J.. Symplectic and Poisson structures of certain moduli spaces (preprint).Google Scholar
[9]Huebschmann, J.. Poisson geometry of certain moduli spaces. International Geometrical Colloquium, Moscow, 1993 (to appear).Google Scholar
[10]Kobayashi, S. and Nomizu, K.. Foundations of differential geometry, I (Interscience, 1963).Google Scholar
[11]Raghunatan, M. S.. Discrete subgroups of Lie groups (Springer, 1972).CrossRefGoogle Scholar