Towards a monopole Fueter Floer homology I: a compactness theorem

Motivated by a conjecture of Donaldson and Segal, we take a first step towards defining a new 3-manifold Floer theory, where the complex is defined by a count of Fueter sections of a hyperk\"ahler bundle over the 3-manifold with fibers modeled on the moduli space of centered monopoles on $\mathbb{R}^3$ with charge $k \in \mathbb{Z}_{\geq 2}$. The main difficulty in defining these counts comes from the non-compactness problems. In this writing, we prove a compactness theorem in this direction in the case of $k=2$.