Poisson structures on certain moduli spaces for bundles on a surface

Huebschmann, Johannes

Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 65-91.

Soient $\Sigma$ une surface fermée, $G$ un groupe de Lie compact, avec algèbre de Lie $g$, et $\xi :P\to \Sigma$ un $G$-fibré principal. Dans des travaux antérieurs nous avons démontré que l’espace des modules $N\left(\xi \right)$ 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 $N\left(\xi \right)$ sur un certain espace de représentations ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$ qui est un difféomorphisme par rapport à des structures adaptées lisses $C^{\infty }\left(N\right(\xi \left)\right)$ et $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$, $\Gamma$ étant l’extension centrale universelle du groupe fondamental de $\Sigma$. Etant donnée une forme symétrique invariante sur $g^{*}$, nous construisons ici des structures de Poisson sur $C^{\infty }\left(N\right(\xi \left)\right)$ et $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$ de sorte que le difféomorphisme mentionné soit compatible avec ces structures. Si la forme sur $g^{*}$ est non-dégénérée, l’espace ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$ 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 $\Sigma$.

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi :P\to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N\left(\xi \right)$ 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 $N\left(\xi \right)$ onto a certain representation space ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty }\left(N\right(\xi \left)\right)$ and $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$, 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 $C^{\infty }\left(N\right(\xi \left)\right)$ and $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$ 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 ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$ 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 $\Sigma$.

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