@articleHuebschmann1995,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 = ,volume = 45,year = 1995,
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TY - JOURAU - Huebschmann, JohannesTI - Poisson structures on certain moduli spaces for bundles on a surfaceJO - Annales de l'institut FourierPY - 1995PB - Association des Annales de l'Institut FourierVL - 45IS - 1SP - 65EP - 91AB - 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 - engKW - 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 bundlesUR - ER -
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 2ff7e9595c
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