Seiberg-Witten monopoles with multiple spinors on a surface times a circle

The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in $\mathcal{M}$ and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing $\mathcal{M}$ in terms of holomorphic data over the surface. We define analytic and algebro-geometric compactifications of $\mathcal{M}$, and construct a homeomorphism between them. For a generic choice of circle-invariant parameters of the equation, Fueter sections do not appear and $\mathcal{M}$ is a compact K\"ahler manifold. After a perturbation it splits into isolated points which can be counted with signs, yielding a number independent of the initial choice of the parameters. We compute this number for surfaces of low genus.