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

Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $\tau$. 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 $\tau(D) = D$, this restricts to an involution on the moduli space $M(r,D)$ of semistable Higgs bundles of rank $r$ with fixed determinant $\mathcal{O}(D)$ and trace-free Higgs field. The fixed point sets of these involutions $M(r,d)^{\tau}$ and $M(r,D)^{\tau}$ are $(A,A,B)$-branes introduced by Baraglia-Schaposnik. In this paper, we derive formulas for the mod 2 Betti numbers of $M(r,d)^{\tau}$ and $M(r,D)^{\tau}$ when $r=2$ and $d$ is odd. In the course of this calculation, we also compute the mod 2 cohomology ring of $SP^m(X)^{\tau}$, the fixed point set of the involution induced by $\tau$ on symmetric products of the Riemann surface.