Moduli spaces of vector bundles with fixed determinant over a real curve

Abstract

Let $(Σ,τ)$ denote a Riemann surface of genus $g \geq 2$ equipped with an anti-holomorphic involution $τ$. In this paper we study the topology of the moduli space $M(r,ξ)^τ$ of stable Real vector bundles over $(Σ,τ)$ of rank $r$ and fixed determinant $ξ$ of degree coprime to $r$. We prove that $M(r,ξ)^τ$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $M(r,ξ)$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the mod $2$ Betti numbers of $M(r,ξ)^τ$ and compute mod $p$ Betti numbers for odd $p$ through a range of degrees. We deduce that if $r$ is even and $ g >>0$, then $M(r,ξ)^τ$ and $M(r,ξ')^τ$ have non-isomorphic cohomology groups unless $ξ$ and $ξ'$ have equivalent Stieffel-Whitney classes modulo automorphisms of $(Σ,τ)$. If $r$ is even, and $g>>0$ is even, we prove that the Betti numbers of $M(r,ξ)^τ$ distinguish topological types of $(Σ, τ; ξ)$. If $r=2$ and $g$ is odd, we compute all mod $p$ Betti numbers of $M(2,ξ)^τ$. MR 32L05, 14P25.