The ASD moduli-space construction conjecture for the bounding cochain

Let $(M,\mathcal P_M)$ be as in Theorem 61, let $\Sigma$ be the cylindrical end of $M$, and let $R(M;\mathcal P_M)$ and $R(\Sigma;\mathcal P_\Sigma)$ denote the corresponding representation spaces. Assume Assumption 61 and that $R(M;\mathcal P_M)$ has transversal self-intersection. For the moduli space $\mathcal M(M\times\mathbb R_\tau;\mathcal P_M\times\mathbb R_\tau;E)$ of finite-energy anti-self-dual connections modulo translation in the $\tau$-direction, one can choose an $\mathbb R_\tau$-invariant perturbation supported on a compact subset of $M$. ASD moduli-space construction conjecture. For this perturbation, the moduli space is a finite-dimensional manifold, admits a compactification whose singular locus has codimension $4$, and every element is represented by a connection $A$ with limits $[a],[b]\in R(M;\mathcal P_M)$ as $\tau\to-\infty$ and $\tau\to+\infty$, respectively, together with a class $[\alpha]\in R(\Sigma;\mathcal P_\Sigma)$ independent of $\tau$ such that

$$\lim_{t\to\infty}A\vert_{\Sigma\times\{(t,\tau)\}}=\alpha$$

for every $\tau$, so that the restrictions of $a$ and $b$ to $\Sigma$ are gauge equivalent to $\alpha$. If the moduli space has dimension zero, then $[a]\ne[b]$. Defining

$${\rm Switch}=\{([a],[b])\mid [a]\ne[b],\ [a\vert_\Sigma]=[b\vert_\Sigma]\},$$

and letting $c([a],[b])$ count the elements in the zero-dimensional component with these asymptotic limits, the bounding cochain is

$$b_M=\sum_{([a],[b])\in{\rm Switch}}c([a],[b])([a],[b])\in CF(R(M;\mathcal P_M),R(M;\mathcal P_M)).$$

The conjecture would provide a direct construction of the bounding cochain $b_M$ by counting finite-energy ASD connections. The author states that it is currently unproved and appears difficult; the theorem containing $b_M$ is instead proved by a different method in later subsections.