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

Abstract

Consider a Riemann surface $X$ of genus $g\ge 2$ 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 $S{P}^m(X{)}^{\tau }$, the fixed point set of the involution induced by $\tau$ on symmetric products of the Riemann surface.

arXiv commentMinor corrections. 21 pages. Comments welcome