Poisson structures on certain moduli spaces for bundles on a surface

Abstract

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \colon P \to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi)$ of central Yang- Mills connections, for appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a diffeomorphism from $N(\xi)$ onto a certain representation space $\roman{Rep}_{\xi}(\Gamma,G)$, with reference to suitable smooth structures $C^{\infty}(N(\xi))$ and $C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ where $\Gamma$ denotes the universal central extension of the fundamental group of $\Sigma$. Given an invariant symmetric bilinear form on $g^*$, we construct here Poisson structures on $C^{\infty}(N(\xi))$ and $C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ 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 $\roman{Rep}_{\xi}(\Gamma,G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a {\it stratified symplectic space\/}, preserved by the induced action of the mapping class group of $\Sigma$.