The singularities of Yang-Mills connections for bundles on a surface. I. The local model

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 $$d_A\ast K_A=0$$ are defined, where $K_A$ refers to the curvature of a connection $A$. For every central Yang-Mills connection $A$, the data induce a structure of unitary representation of the stabilizer $Z_A$ on the first cohomology group $\text{roman}{H}_A^1(\mathrm{\Sigma },\text{roman}ad(\xi ))$ with coefficients in the adjoint bundle $\text{roman}ad(\xi )$, with reference to $A$, with momentum mapping ${\mathrm{\Theta }}_A$ from $\text{roman}{H}_A^1(\mathrm{\Sigma },\text{roman}ad(\xi ))$ to the dual $z_A^{\ast }$ of the Lie algebra $z_A$ of $Z_A$. We show that, for every central Yang-Mills connection $A$, a suitable Kuranishi map identifies a neighborhood of zero in the Marsden-Weinstein reduced space $\text{roman}{H}_A$ for ${\mathrm{\Theta }}_A$ with a neighborhood of the point $[A]$ in the moduli space of central Yang-Mills connections on $\xi$.