2026/07/22 by Florent Schaffhauser, Tommaso Scognamiglio
Mathematics · #math.AG #math.DG
Let X be a Riemann surface of genus g \geqslant 2 and let σ: X → X be an antiholomorphic involution on X. Let 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 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 ℝN(r,d) is still equal to that of ℝPicd. In contrast, when the base curve has empty real locus and r and d are not coprime, the number of connected components of ℝN(r,d) can be smaller than that of ℝPicd. 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.