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

TL;DR

This paper investigates the local structure of the moduli space of Yang-Mills connections on a surface, using Kuranishi maps and symplectic reduction to understand singularities at central solutions.

Contribution

It introduces a local model for the moduli space near central Yang-Mills connections using Kuranishi maps and Marsden-Weinstein reduction techniques.

Findings

Identifies neighborhoods of central Yang-Mills connections with reduced spaces.

Describes the action of the stabilizer group on cohomology.

Provides a framework for analyzing singularities in the moduli space.

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