Fueter sections and $\mathbb{Z}_2$-harmonic 1-forms

TL;DR

This paper proves a compactness theorem for Fueter sections of charge 2 monopole bundles over 3-manifolds, showing that diverging sequences converge to non-zero $ ext{Z}_2$-harmonic 1-forms, motivated by conjectures in Calabi-Yau geometry.

Contribution

It establishes a new compactness result for Fueter sections, linking monopole bundle sequences to $ ext{Z}_2$-harmonic 1-forms, advancing understanding in geometric analysis.

Findings

Diverging Fueter sections, after renormalization, converge to $ ext{Z}_2$-harmonic 1-forms.

The convergence occurs in the $W^{1,2}$-topology.

Supports conjectures relating monopoles and special Lagrangians in Calabi-Yau 3-folds.

Abstract

Motivated by a conjecture of Donaldson and Segal on the counts of monopoles and special Lagrangians in Calabi-Yau 3-folds, we prove a compactness theorem for Fueter sections of charge 2 monopole bundles over 3-manifolds: Let $u_k$ be a sequence of Fueter sections of the charge 2 monopole bundle over a closed oriented Riemannian 3-manifold $(M,g)$, with $L^\infty$-norm diverging to infinity. Then a renormalized sequence derived from $u_k$ subsequentially converges to a non-zero $\mathbb{Z}_2$-harmonic 1-form $\mathcal{V}$ on $M$ in the $W^{1,2}$-topology.