arXiv research

Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.

Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.

We prove that a sequence of solutions of the Seiberg-Witten equation with multiple spinors in dimension three can degenerate only by converging (after rescaling) to a Fueter section of a bundle of moduli spaces of ASD instantons.

I construct multiplicies and orientations of tangent cones to any blow-up set $Z$ for the Seiberg-Witten equation with multiple spinors. This is used to prove that $Z$ determines a homology class, which is shown to be equal to the Poincaré dual of the first Chern class of the determinant line bundle. I also obtain a lo…

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 sign…

We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan-Warner equations.…

We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furt…

Compactness theorem for 3D RS-SW equations on 3-manifolds.

New method constructs multi-monopoles on mapping tori.