Existence of multi-monopoles on mapping tori

TL;DR

This paper constructs explicit examples of multi-monopoles on mapping tori, revealing their wall-crossing behavior and non-invariance in counts, advancing understanding of multi-spinor gauge theories.

Contribution

It proves an adiabatic limit theorem that constructs multi-monopoles on mapping tori, enabling explicit examples and analysis of wall-crossing phenomena.

Findings

Constructed multi-monopoles on non-product 3-manifolds.

Demonstrated wall-crossing behavior in multi-monopole counts.

Provided the first explicit examples of multi-monopoles in various chambers.

Abstract

While the Seiberg-Witten equations have been well-studied on 3-manifolds, their multiple spinor generalisation exhibits some unexpected behaviour. Most notably, the count of these "multi-monopoles" does not define a topological invariant. Instead, the count can jump as parameters of the equations cross between certain regions in the parameter space, known as chambers. This wall-crossing phenomenon is related to deep questions about multi-valued harmonic spinors and higher-dimensional gauge theory. However, concrete examples of this behaviour have not been studied, primarily because the existing constructions of multi-monopoles are not rich enough for wall-crossing to be observed. We address this by proving an adiabatic limit theorem, which constructs multi-monopoles for a wide range of parameters on mapping tori. These solutions are obtained by perturbing the fixed points of the monodromy map associated to a family of multi-vortex moduli spaces. We use our theorem to produce the first explicit constructions of multi-monopoles on non-product 3-manifolds in various chambers.