no. 1
Poisson structures on certain moduli spaces for bundles on a surface
Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 65-91.

Soient Σ une surface fermée, G un groupe de Lie compact, avec algèbre de Lie g, et ξ:P→Σ un G-fibré principal. Dans des travaux antérieurs nous avons démontré que l’espace des modules N(ξ) 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 N(ξ) sur un certain espace de représentations Rep ξ (Γ,G) qui est un difféomorphisme par rapport à des structures adaptées lisses C ∞ (N(ξ)) et C ∞ Rep ξ (Γ,G), Γ étant l’extension centrale universelle du groupe fondamental de Σ. Etant donnée une forme symétrique invariante sur g * , nous construisons ici des structures de Poisson sur C ∞ (N(ξ)) et C ∞ Rep ξ (Γ,G) de sorte que le difféomorphisme mentionné soit compatible avec ces structures. Si la forme sur g * est non-dégénérée, l’espace Rep ξ (Γ,G) 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 Σ.

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.

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     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},
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Huebschmann, Johannes. Poisson structures on certain moduli spaces for bundles on a surface. Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 65-91. doi : 10.5802/aif.1448. https://aif.centre-mersenne.org/articles/10.5802/aif.1448/

[1] J. M. Arms, R. Cushman, and M. J. Gotay, A universal reduction procedure for Hamiltonian group actions, in : The geometry of Hamiltonian systems, T. Ratiu, ed. MSRI Publ., 20 (1991), Springer Berlin-Heidelberg-New York-Tokyo, 33-51. | MR | Zbl

[2] M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London, A 308 (1982), 523-615. | MR | Zbl

[3] W. M. Goldman, The symplectic nature of the fundamental groups of surfaces, Advances in Math., 54 (1984), 200-225. | MR | Zbl

[4] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Inventiones Math., 85 (1986), 263-302. | MR | Zbl

[5] J. Huebschmann, Poisson cohomology and quantization, J. für die Reine und Angewandte Mathematik, 408 (1990), 57-113. | MR | Zbl

[6] J. Huebschmann, On the quantization of Poisson algebras, Symplectic Geometry and Mathematical Physics, Actes du colloque en l'honneur de Jean-Marie Souriau, P. Donato, C. Duval, J. Elhadad, G.M. Tuynman, eds. ; Progress in Mathematics, Vol. 99, Birkhäuser, Boston Basel Berlin, (1991), 204-233. | MR | Zbl

[7] J. Huebschmann, The singularities of Yang-Mills connections for bundles on a surface. I. The local model, Math. Z. (to appear). | Zbl

[8] J. Huebschmann, The singularities of Yang-Mills connections for bundles on a surface II. The stratification, Math. Z. (to appear). | Zbl

[9] J. Huebschmann, Holonomies of Yang-Mills connections for bundles on a surface with disconnected structure group, Math. Proc. Cambr. Phil. Soc, 116 (1994), 375-384. | MR | Zbl

[10] J. Huebschmann, Smooth structures on certain moduli spaces for bundles on a surface, preprint 1992. | Zbl

[11] J. Huebschmann, The singularities of Yang-Mills connections for bundles on a surface. III. The identification of the strata, in preparation. | Zbl

[12] J. Huebschmann, Poisson geometry of flat connections for SU(2)-bundles on surfaces, Math. Z. (to appear). | Zbl

[13] J. Huebschmann, Symplectic and Poisson structures of certain moduli spaces, Duke Math. (to appear). | Zbl

[14] J. Huebschmann and L. Jeffrey, Group cohomology construction of symplectic forms on certain moduli spaces, Int. Math. Research Notices, 6 (1994), 245-249. | MR | Zbl

[15] Y. Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc., 116 (1992), 591-605. | MR | Zbl

[16] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetries, Rep. on Math. Phys., 5 (1974), 121-130. | MR | Zbl

[17] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math., 82 (1965), 540-567. | MR | Zbl

[18] M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math., 89 (1969), 19-51. | MR | Zbl

[19] M. S. Narasimhan and S. Ramanan, 2θ-linear systems on abelian varieties, Bombay Colloquium, (1985), 415-427. | MR | Zbl

[20] C. S. Seshadri, Spaces of unitary vector bundles on a compact Riemann surface, Ann. of Math., 85 (1967), 303-336. | MR | Zbl

[21] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math., 134 (1991), 375-422. | MR | Zbl

[22] A. Weinstein, On the symplectic structure of moduli space, A. Floer memorial, Birkhäuser Verlag, to appear. | Zbl

[23] H. Whitney, Analytic extensions of differentiable functions defined on closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. | JFM | MR | Zbl

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