Let be a closed surface, a compact Lie group, with Lie algebra , and a principal -bundle. In earlier work we have shown that the moduli space 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 onto a certain representation space , in fact a diffeomorphism, with reference to suitable smooth structures and , where denotes the universal central extension of the fundamental group of . Given a coadjoint action invariant symmetric bilinear form on , we construct here Poisson structures on and in such a way that the mentioned diffeomorphism identifies them. When the form on is non-degenerate the Poisson structures are compatible with the stratifications where 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 .
Soient une surface fermée, un groupe de Lie compact, avec algèbre de Lie , et un -fibré principal. Dans des travaux antérieurs nous avons démontré que l’espace des modules de connexions centrales de Yang-Mills, par rapport à des données adaptées supplémentaires, est stratifié par des variétés symplectiques et que l’holonomie fournit un homéomorphisme de sur un certain espace de représentations qui est un difféomorphisme par rapport à des structures adaptées lisses et , étant l’extension centrale universelle du groupe fondamental de . Etant donnée une forme symétrique invariante sur , nous construisons ici des structures de Poisson sur et de sorte que le difféomorphisme mentionné soit compatible avec ces structures. Si la forme sur est non-dégénérée, l’espace étant muni de la stratification correspondante, ces structures de Poisson sont compatibles avec les stratifications et fournissent donc des structures d’espaces symplectiques stratifiés, conservées par l’action du groupe des classes d’applications de .
@article{AIF_1995__45_1_65_0, author = {Huebschmann, Johannes}, title = {Poisson structures on certain moduli spaces for bundles on a surface}, journal = {Annales de l'Institut Fourier}, pages = {65--91}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {45}, number = {1}, year = {1995}, doi = {10.5802/aif.1448}, zbl = {0819.58010}, mrnumber = {96a:58038}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1448/} }
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 SP - 65 EP - 91 VL - 45 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1448/ DO - 10.5802/aif.1448 LA - en ID - AIF_1995__45_1_65_0 ER -
%0 Journal Article %A Huebschmann, Johannes %T Poisson structures on certain moduli spaces for bundles on a surface %J Annales de l'Institut Fourier %D 1995 %P 65-91 %V 45 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1448/ %R 10.5802/aif.1448 %G en %F AIF_1995__45_1_65_0
Huebschmann, Johannes. Poisson structures on certain moduli spaces for bundles on a surface. Annales de l'Institut Fourier, Volume 45 (1995) no. 1, pp. 65-91. doi : 10.5802/aif.1448. https://aif.centre-mersenne.org/articles/10.5802/aif.1448/
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