Poisson structures on certain moduli spaces for bundles on a surface

Johannes Huebschmann

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 1, page 65-91
  • ISSN: 0373-0956

Abstract

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Let Σ be a closed surface, G a compact Lie group, with Lie algebra g , and ξ : P → Σ a principal G -bundle. In earlier work we have shown that the moduli space N ( ξ ) of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from N ( ξ ) onto a certain representation space Rep ξ ( Γ , G ) , in fact a diffeomorphism, with reference to suitable smooth structures C ∞ ( N ( ξ ) ) and C ∞ Rep ξ ( Γ , G ) , where Γ denotes the universal central extension of the fundamental group of Σ . Given a coadjoint action invariant symmetric bilinear form on g * , we construct here Poisson structures on C ∞ ( N ( ξ ) ) and C ∞ Rep ξ ( Γ , G ) in such a way that the mentioned diffeomorphism identifies them. When the form on g * is non-degenerate the Poisson structures are compatible with the stratifications where Rep ξ ( Γ , G ) is endowed with the corresponding stratification and, furthermore, yield structures of a stratified symplectic space, preserved by the induced action of the mapping class group of Σ .

How to cite

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Huebschmann, Johannes. "Poisson structures on certain moduli spaces for bundles on a surface." Annales de l'institut Fourier 45.1 (1995): 65-91. <http://eudml.org/doc/75119>.

@article{Huebschmann1995,
abstract = {Let $\Sigma $ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \colon P \rightarrow \Sigma $ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi )$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N(\xi )$ onto a certain representation space $\{\rm Rep\}_\{\xi \}(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^\{\infty \}(N(\xi ))$ and $C^\{\infty \}\left(\{\rm Rep\}_\{\xi \}(\Gamma ,G)\right)$, where $\Gamma $ denotes the universal central extension of the fundamental group of $\Sigma $. Given a coadjoint action invariant symmetric bilinear form on $g^*$, we construct here Poisson structures on $C^\{\infty \}(N(\xi ))$ and $C^\{\infty \}\left(\{\rm Rep\}_\{\xi \}(\Gamma ,G)\right)$ in such a way that the mentioned diffeomorphism identifies them. When the form on $g^*$ is non-degenerate the Poisson structures are compatible with the stratifications where $\{\rm Rep\}_\{\xi \}(\Gamma ,G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a stratified symplectic space, preserved by the induced action of the mapping class group of $\Sigma $.},
author = {Huebschmann, Johannes},
journal = {Annales de l'institut Fourier},
keywords = {geometry of principal bundles; singularities of smooth mappings; symplectic reduction with singularities; Yang-Mills connections; stratified symplectic space; Poisson structure; geometry of moduli spaces; representation spaces; categorical quotient; geometric invariant theory; moduli of vector bundles},
language = {eng},
number = {1},
pages = {65-91},
publisher = {Association des Annales de l'Institut Fourier},
title = {Poisson structures on certain moduli spaces for bundles on a surface},
url = {http://eudml.org/doc/75119},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Huebschmann, Johannes
TI - Poisson structures on certain moduli spaces for bundles on a surface
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 1
SP - 65
EP - 91
AB - Let $\Sigma $ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \colon P \rightarrow \Sigma $ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi )$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N(\xi )$ onto a certain representation space ${\rm Rep}_{\xi }(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty }(N(\xi ))$ and $C^{\infty }\left({\rm Rep}_{\xi }(\Gamma ,G)\right)$, where $\Gamma $ denotes the universal central extension of the fundamental group of $\Sigma $. Given a coadjoint action invariant symmetric bilinear form on $g^*$, we construct here Poisson structures on $C^{\infty }(N(\xi ))$ and $C^{\infty }\left({\rm Rep}_{\xi }(\Gamma ,G)\right)$ in such a way that the mentioned diffeomorphism identifies them. When the form on $g^*$ is non-degenerate the Poisson structures are compatible with the stratifications where ${\rm Rep}_{\xi }(\Gamma ,G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a stratified symplectic space, preserved by the induced action of the mapping class group of $\Sigma $.
LA - eng
KW - geometry of principal bundles; singularities of smooth mappings; symplectic reduction with singularities; Yang-Mills connections; stratified symplectic space; Poisson structure; geometry of moduli spaces; representation spaces; categorical quotient; geometric invariant theory; moduli of vector bundles
UR - http://eudml.org/doc/75119
ER -

References

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