Existence and chamber invariance conjecture for Seiberg–Witten $U(2)$ monopole Floer homology

Suppose that $Y$ is a closed oriented $3$-manifold, and $E$ is a $U(2)$ bundle with admissibility condition $w=c_1(E)\in H^2(Y;\mathbb{Z})$ a non-torsion odd class. Choose a spin-$c$ structure $\mathfrak{s}$, let $(S,\rho)$ be a spinor bundle representing it, and set $V=S\otimes E$, representing the spin-$u$ structure $\mathfrak{t}$. For $\tau\neq\pi,2\pi n$ with $n\in\mathbb{Z}$, consider the Seiberg–Witten $U(2)$ monopole Floer homology $HM_{U(2)}(Y,\mathfrak{t},\tau)$. Seiberg–Witten $U(2)$ monopole Floer homology conjecture. It should be possible to define these groups with the following features: the chain complex is built from a suitably perturbed version of the $U(2)$ monopole equations; its differential counts gradient flow lines modulo $\mathbb{R}$-translations of a perturbed $U(2)$ Chern–Simons–Dirac functional; Novikov coefficients are generally required because of monotonicity issues; the groups are generally $\mathbb{Z}/2\mathbb{Z}$-graded; and, if

$$\mathbb{R}\backslash\{\pi,2\pi n\mid n\in\mathbb{Z}\}=\bigcup_i I_i$$

is the decomposition into disjoint open intervals, then $HM_{U(2)}(Y,\mathfrak{t},\tau)\simeq HM_{U(2)}(Y,\mathfrak{t},\tau')$ whenever $\tau$ and $\tau'$ lie in the same interval $I_i$. This proposes a $U(2)$ monopole Floer theory for admissible bundles and predicts invariance within chambers, while leaving its analytic construction and the stated properties open.