Smooth structures on certain moduli spaces for bundles on a surface

TL;DR

This paper constructs smooth structures on certain moduli spaces of bundles and connections on a surface, establishing a diffeomorphism with representation spaces and analyzing their infinitesimal geometry, revealing differences from standard structures.

Contribution

It introduces explicit smooth structures on moduli spaces of Yang-Mills connections and representation spaces, and relates them via holonomy maps, with detailed geometric analysis.

Findings

Diffeomorphism between moduli space and representation space via holonomy

Smooth structures differ from standard on genus two surface

Infinitesimal geometry characterized by twisted integration mapping

Abstract

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, $\xi \colon P \to \Sigma$ a principal $G$-bundle, let $N(\xi)$ denote the moduli space of central Yang-Mills connections on $\xi$, for suitably chosen additional data, and let $\roman{Rep}_{\xi}(\Gamma,G)$ be the space of representations of the universal central extension $\Gamma$ of the fundamental group of $\Sigma$ in $G$ that corresponds to $\xi$. We construct smooth structures on $N(\xi)$ and $\roman{Rep}_{\xi}(\Gamma,G)$ and show that the assignment to a smooth connection $A$ of its holonomies with reference to suitable closed paths yields a diffeomorphism from $N(\xi)$ onto $\roman{Rep}_{\xi}(\Gamma,G)$; moreover we show that the derivative of the latter at the non-singular points of $N(\xi)$ amounts to a certain twisted integration mapping. Finally we examine the infinitesimal geometry of these moduli spaces with reference to the smooth structures and, for illustration, we show that, on the moduli space of flat $\roman{SU}(2)$-connections for a surface of genus two which, as a space, is just complex projective 3-space, our smooth structure looks rather different from the standard structure.