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

Abstract

Abstract 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 S y m m ( X ) τ , the fixed point set of the involution induced by τ on symmetric products of the Riemann surface.