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

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 $L^{\mathrm{\infty }}$-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.