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

Abstract

Let $Σ$ 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 $ξ\colon P \to Σ$ be a principal $G$-bundle, and pick a Riemannian metric and orientation on $Σ$, so that the corresponding Yang-Mills equations $$d_A*K_A = 0$$ 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 $\roman H^1_A(Σ,\roman{ad}(ξ))$ with coefficients in the adjoint bundle $\roman{ad}(ξ)$, with reference to $A$, with momentum mapping $Θ_A$ from $\roman H^1_A(Σ,\roman{ad}(ξ))$ to the dual $z^*_A$ 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 $\roman H_A$ for $Θ_A$ with a neighborhood of the point $[A]$ in the moduli space of central Yang-Mills connections on $ξ$.