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

Abstract

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

31 pages. I added a conclusion to Theorem 1.2 that provides better context for Theorem 1.3