From what I understand about work on the Witten conjecture relating Donaldson and Seiberg-Witten invariants, the main strategy has been to relate them with the use of the "SO(3) monopole" theory developed by Feehan and Leness (see their paper here).

Both the Donaldson invariants and the Seiberg-Witten invariants have closely related Floer homologies for $3$-manifolds.

Is it possible to define such a Floer homology for $SO(3)$ monopoles? Are there clear reasons that this may or may not be possible?

As a first step, I am curious about whether there is a natural Chern-Simons-like functional $\mathcal{L}$ such that $SO(3)$ monopoles on a cylinder $\mathbb{R} \times Y$ correspond to negative gradient flowlines of $\mathcal{L}$.


It seems that not much work has been done for $SO(3)$ monopoles on three-manifolds. As far as I know, nobody has yet defined a Floer homology for that. However, Feehan, Leness and Kutluhan have started to investigate that, see section 3 of https://www.math.ias.edu/files/feehanreport.pdf . They don't seem to be interested in constructing an $SO(3)$ monopole homology, but they have worked on some issues crucial to developing such a theory (finding appropriate perturbations, solving compactness issues and studying the structure of the flow lines).

They say that $SO(3)$ monopoles on $\mathbb{R} \times Y$ correspond to the gradient flow lines of the $U(2)$ Chern-Simons-Dirac functional. If I am not mistaken, it is just the $U(2)$ Chern-Simons functional plus the usual spinor contribution (pairing between $\Phi$ and $D_B \Phi$).


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