Solutions to the Seiberg-Witten equations in all dimensions

Abstract

This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented $n$ dimensional Riemannian manifold with a spin$^\mathbb{C}$-structure, we described an elliptic system of equations that recovers the traditional Seiberg-Witten equations in dimensions $3$ and $4$. The paper focuses on constructing explicit solutions of these equations in dimensions $5, 6$ and $8$, where harmonic perturbation terms are sometimes required to ensure solutions. In dimensions $6$ and $8$ we construct solutions on Kähler manifolds and relate these solutions to vortices. In dimension $5$, we construct solutions on the product of a closed Riemann surface and $\mathbb{R}^3$. The solutions are invariant in the $\mathbb{R}^3$ directions and can be related to vortices on the Riemann surface. A key issue in higher dimensions is the potential noncompactness of the space of solutions, in contrast to the compact moduli spaces in lower dimensions. In our solutions, this noncompactness is linked to the presence of certain odd-dimensional harmonic forms, with an explicit example provided in dimension $6$.