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

Abstract

Let $\Sigma$ 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 $\xi :P\to \Sigma$ be a principal $G$-bundle, and pick a Riemannian metric and orientation on $\Sigma$, 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 $\roman H^1_A(Σ,\roman{ad}(ξ))$ with coefficients in the adjoint bundle $\roman{ad}(ξ)$, with reference to $A$, with momentum mapping ${\Theta }_A$ from $\roman H^1_A(Σ,\roman{ad}(ξ))$ 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 $\roman H_A$ for ${\Theta }_A$ with a neighborhood of the point $[A]$ in the moduli space of central Yang-Mills connections on $\xi$.

14 pages

References in corpus (1)

• Symplectic and Poisson structures of certain moduli spaces

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