Towards a monopole Fueter Floer homology I: a compactness theorem

Abstract

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 hyperkähler bundle over the 3-manifold with fibers modeled on the moduli space of centered monopoles on ${\mathbb{R}}^3$ with charge $k\in {\mathbb{Z}}_{\ge 2}$. 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 $k=2$.

30 pages