The symplectic embedding–configuration space duality conjecture for up to eight balls

Let $n\leq 8$ and let ${\mathbf c}=c_1\geq\cdots\geq c_n>0$ be admissible capacities. Write $\Im\!\operatorname{Emb}_n({\mathbf c},{\mathbf C}{\mathbf P}^2)$ for the space of symplectic embeddings of $n$ disjoint balls with capacities ${\mathbf c}$ into ${\mathbf C}{\mathbf P}^2$, and let $\operatorname{Conf}_n({\mathbf C}{\mathbf P}^2)$ denote the configuration space of $n$ distinct points. The symplectic embedding–configuration space duality conjecture. The space

$$\Im\!\operatorname{Emb}_n({\mathbf c},{\mathbf C}{\mathbf P}^2)$$

is homotopy equivalent to a union of strata in $\operatorname{Conf}_n({\mathbf C}{\mathbf P}^2)$ defined by the relative positions of the $n$ points with respect to generic immersed holomorphic spheres in ${\mathbf C}{\mathbf P}^2$. This conjecture proposes that the duality between genericity conditions for point configurations and numerical conditions on ball capacities, established in the paper for $n\leq 4$, continues through $n=8$; the cases $n\geq 5$ remain open.