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

Abstract

Let $X$ be a Riemann surface of genus $g⩾2$ 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.

21 pages, 1 figure, All comments and questions are welcome!