Moduli varieties of real and quaternionic bundles over a curve

We examine the moduli problem for real and quaternionic vector bundles over a curve, and we give a gauge-theoretic construction of would-be moduli varieties for such bundles. These moduli varieties are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study Gal(C/R)-actions on moduli varieties of semistable holomorphic bundles over a complex curve with a given real structure. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of those actions by $2^g +1$, where g is the genus of the curve. Moreover, we show that any two such connected components are homeomorphic.