Connected components of real loci in moduli spaces of vector and Higgs bundles over a Klein surface

Let $X$ be a Riemann surface of genus $g \geqslant 2$ and let $\sigma : X \to X$ be an antiholomorphic involution on $X$.

Let $\mathcal{N}(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ on $X$, with the induced real structure.

Using a gauge-theoretic approach, we determine the number of connected components of the real locus of $\mathcal{N}(r,d)$ for general $r$ and $d$.

We show in particular that, when the base curve has real points, quaternionic vector bundles can exist for even rank and degree but that the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ is still equal to that of $\mathbb{R}\mathrm{Pic}_d$.

In contrast, when the base curve has empty real locus and $r$ and $d$ are not coprime, the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ can be smaller than that of $\mathbb{R}\mathrm{Pic}_d$.

We then generalize these results to real loci of moduli spaces of Higgs bundles and apply them to the study of the topology of certain $(A,A,A)$ and $(A,B,A)$ branes in the associated hyperkähler quotient.