Del Pezzo–cyclic loop space duality conjecture

Let $\mathbb{B}_k$, for $0 \leq k \leq 8$, and $\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ be the indicated series of del Pezzo surfaces. Let $\mathcal{L}^{k}_c S^4$, for $0 \leq k \leq 8$, be the corresponding series of iterated loop spaces, and let $IIB$ denote the topological model referred to in the source. Del Pezzo–cyclic loop space duality conjecture. There must be an explicit relation between the series of del Pezzo surfaces and the series of iterated loop spaces. In particular, blowing up a del Pezzo surface should correspond to taking a cyclification of an iterated cyclification of $S^4$. This relation should match the $E_k$ symmetry patterns occurring in both series and relate other geometric data, such as the volumes of curves on del Pezzo surfaces with certain metric data on the iterated loop spaces. The conjecture proposes a mathematical duality between the algebraic geometry of del Pezzo surfaces and the algebraic topology of cyclic loop spaces of the $4$-sphere; the precise relation and its geometric consequences remain open.