Mysterious Triality

Classical del Pezzo surfaces exhibit an unexpected connection with exceptional E‑type root systems, arising from the intersection form on their homology. In 2001, the mathematical physicists Iqbal, Neitzke, and Vafa observed an analogous appearance of E-type root systems in the study of 1/2-BPS branes in M-theory compactifications, such as type IIA string theory. At the time, the link to del Pezzo surfaces remained obscure, prompting them to dub this correspondence Mysterious Duality.

In a joint work with Hisham Sati 20+ years later, we observed an analogous structure in a purely topological setting, namely in the toroidifications

T^k S^4:=Map(T^k,S4^)//T^k,         0≤k≤8,

defined as homotopy quotients of mapping spaces from tori to the 4‑sphere. We showed that the rational homotopy models of these spaces naturally give rise to root systems of type E_k, via symmetries realized by rational self‑equivalences. We also showed that these spaces serve as universal target spaces for compactified M-theory and as classifying spaces for supergravity fields. This reveals a new duality—no less enigmatic—between del Pezzo surfaces and toroidifications of S^4, culminating in Mysterious Triality, which intertwines Geometry, Topology, and Physics.