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

## TL;DR

This paper investigates the topology of moduli spaces of stable Real vector bundles over real curves, revealing their Lagrangian structure, computing Betti numbers, and distinguishing topological types based on genus and rank.

## Contribution

It establishes the Lagrangian and monotone properties of these moduli spaces and provides recursive formulas and explicit Betti number computations for various cases.

## Key findings

- Moduli spaces are orientable, monotone Lagrangian submanifolds.
- Recursive formulas for mod 2 Betti numbers are derived.
- Betti numbers distinguish topological types of real curves and vector bundles.

## Abstract

Let $(\Sigma,\tau)$ denote a Riemann surface of genus $g \geq 2$ 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 $M(r,\xi')^{\tau}$ have non-isomorphic cohomology groups unless $\xi$ and $\xi'$ 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.