Smooth structures on certain moduli spaces for bundles on a surface Original Research Article

Let Σ be a closed surface, G a compact Lie group, not necessarily connected, with Lie algebra image, ξ: P → Σ a principal G-bundle, let N(ξ) denote the moduli space of central Yang-Mills connections on ξ, with reference to suitably chosen additional data, and let Repξ(Γ, G) be the space of representations of the universal central extension Γ of the fundamental group of Σ in G that corresponds to ξ. We construct smooth structures on N(ξ) and Repgx(Γ, G), that is, algebras of continuous functions which restrict to smooth functions on the strata of certain associated stratifications; by means of a detailed investigation of the derivative of the holonomy we show thereafter that, with reference to these smooth structures, the assignment to a smooth connection A of its holonomies with reference to suitable closed paths yields a diffeomorphism from N(ξ) onto Repξ(Γ, G); moreover, we show that the derivative of the latter at the non-singular points of N(ξ) amounts to a certain twisted integration mapping relating a suitable de Rham theory with group cohomology with appropriate coefficients. Finally, we examine the infinitesimal geometry of these moduli spaces by means of the smooth structures and, for illustration, we show that, on the moduli space of flat 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.