Symplectic and Poisson structures of certain moduli spaces

Abstract

Symplectic and Poisson structures of certain moduli spaces/Huebschmann,J./ Abstract: Let $\pi$ be the fundamental group of a closed surface and $G$ a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction due to A. Weinstein relying on techniques from equivariant cohomology may be refined so as to yield (i) a symplectic structure on a certain smooth manifold $\Cal M(\Cal P,G)$ containing the space $\roman{Hom}(\pi,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M(\Cal P,G)$ preserving the symplectic structure, with momentum mapping $\mu \colon \Cal M(\Cal P,G) \to g^*$, in such a way that the reduced space equals the space $\roman{Rep}(\pi,G)$ of representations. Our approach is somewhat more general in that it also applies to twisted moduli spaces; in particular, it yields the {\smc Narasimhan-Seshadri} moduli spaces of semistable holomorphic vector bundles by {\it symplectic reduction in finite dimensions}.This implies that, when the group $G$ is compact, such a twisted moduli space inherits a structure of {\it stratified symplectic space}, and that the strata of these twisted moduli spaces have finite symplectic volume.