The moduli space of multi-monopoles on a Riemann surface

TL;DR

This paper investigates the moduli space of solutions to Seiberg-Witten equations with multiple spinors on a compact Riemann surface, connecting geometric analysis with algebraic geometry and topological invariants.

Contribution

It provides the first computation of the Euler characteristic and rational homology of these moduli spaces, advancing understanding of their geometric and topological properties.

Findings

Computed the Euler characteristic of the moduli spaces.

Determined the rational homology using spectral curves.

Connected the moduli spaces to algebraic geometry and 3-manifold invariants.

Abstract

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent interest in the geometry of algebraic curves, as they parameterize generalized divisors in Brill-Noether theory for higher rank vector bundles. We compute the Euler characteristic of these spaces, completing a computation initiated by Doan, and then compute their rational homology using spectral curves and techniques of Fulton and Lazarsfeld.