The singularities of Yang-Mills connections for bundles on a surface. II. The stratification

Let $\mathrm{\Sigma }$ be a closed surface, $G$ a compact Lie group, not necessarily connected, with Lie algebra $g$, endowed with an adjoint action invariant scalar product, let $\xi \text{\hspace{0.17em}⁣}:P\to \mathrm{\Sigma }$ be a principal $G$-bundle, and pick a Riemannian metric and orientation on $\mathrm{\Sigma }$ so that the corresponding Yang-Mills equations are defined. In an earlier paper we determined the local structure of the moduli space $N(\xi )$ of central Yang-Mills connections on $\xi$ near an arbitrary point. Here we show that the decomposition of $N(\xi )$ into connected components of orbit types of central Yang-Mills connections is a stratification in the strong (i.~e. Whitney) sense; furthermore each stratum, being a smooth manifold, inherits a finite volume symplectic structure from the given data. This complements, in a way, results of {\smc Atiyah-Bott} in that it will in general decompose further the critical sets of the corresponding Yang-Mills functional into smooth pieces.