Symmetric products of a real curve and the moduli space of Higgs bundles

Consider a Riemann surface X of genus g≥2 equipped with an antiholomorphic involution τ. This induces a natural involution on the moduli space M(r,d) of semistable Higgs bundles of rank r and degree d. If D is a divisor such that τ(D)=D, this restricts to an involution on the moduli space M(r,D) of those Higgs bundles with fixed determinant O(D) and trace-free Higgs field. The fixed point sets of these involutions M(r,d)τ and M(r,D)τ are (A,A,B)-branes introduced by Baraglia and Schaposnik (2016). In this paper, we derive formulas for the mod 2 Betti numbers of M(r,d)τ and M(r,D)τ when r=2 and d is odd. In the course of this calculation, we also compute the mod 2 cohomology ring of Symm(X)τ, the fixed point set of the involution induced by τ on symmetric products of the Riemann surface.