March 16, 2023
TITLE: Towards a Monopole Fueter Floer Homology
ABSTRACT: Monopoles appear as the dimensional reduction of instantons to 3-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. Monopoles, conjecturally, can be used to define invariants of 3-manifolds, Calabi-Yau 3-folds, and G2-manifolds. These monopole invariants, conjecturally, are related to certain counts of calibrated submanifolds, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.
Motivated by this conjecture, we propose a Floer theory for 3-manifolds, generated by Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3. A major difficulty in defining these homology groups 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.