Brad Wilson | Searcharxiv

arXiv2026

Multi-monopoles and monodromy around the wobbly divisor

The multi-monopole equations are a generalisation of the Seiberg--Witten equations where the spinor is coupled to an auxiliary vector bundle. The signed count of multi-monopoles on a 3-manifold can vary according to a new type of wall-crossing phenomenon. For products $S^1\times C$, where $C$ is a closed Riemann surface, we relate the wall and chamber structure to loops in the complement of the wobbly divisor of $C$. Such a loop induces a monodromy map on the multi-vortex moduli space, and the count of its fixed points is expected to agree with the count of multi-monopoles on $S^1\times C$ in the given chamber. By studying singularities of framed multi-vortex moduli spaces, we are able to understand these monodromy maps for various loops. Counting fixed points suggests that product 3-manifolds display incredibly rich wall-crossing behaviour.

math.AG↗

arXiv2026

Existence of multi-monopoles on mapping tori

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.

math.DG↗