In this talk, we propose a monopole Fueter invariant of 3-manifolds, motivated by the Donaldson-Segal program.

The theory of Yang-Mills connections, and in particular instantons, revolutionized the study of 4-manifolds. Monopoles appear as the dimensional reduction of instantons to 3-manifolds. The moduli spaces of monopoles on R3 form ALF hyperkähler manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higher-dimensional spaces. The most interesting examples appear on Calabi-Yau 3-folds and G2-manifolds. Donaldson and Segal hinted at the idea of defining invariants of Calabi-Yau 3-folds by counting monopoles on these manifolds. These monopole invariants, conjecturally, are related to the special Lagrangians, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.

Motivated by this conjecture, we propose numerical invariants of 3-manifolds by counting Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3. More ambitiously, one would hope this would result in a Floer theoretic invariant of 3-manifolds. A major difficulty in defining these invariants is related to the non-compactness problems. We prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.