Poisson geometry of flat connections for $\roman{SU}(2)$-bundles on surfaces

Abstract

In earlier work we have shown that the moduli space $N$ of flat connections for the (trivial) $\roman{SU(2)}$-bundle on a closed surface of genus $\ell \geq 2$ inherits a structure of stratified symplectic space with two connected strata $N_Z$ and $N_{(T)}$ and $2^{2\ell}$ isolated points. In this paper we show that, close to each point of $N_{(T)}$, the space $N$ and its Poisson algebra look like a product of $\bold C^{\ell}$ endowed with the standard symplectic Poisson structure with the reduced space and Poisson algebra of the system of $(\ell-1)$ particles in the plane with total angular momentum zero, while close to one of the isolated points, the Poisson algebra on $N$ looks like that of the reduced system of $\ell$ particles in $\bold R^3$ with total angular momentum zero. Moreover, in the genus two case where the space $N$ is known to be smooth we locally describe the Poisson algebra and the various underlying symplectic structures on the strata and their mutual positions explicitly in terms of the Poisson structure.