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 the negative gradient flow of $ \mathcal{L}$ .