Mysterious Triality

The striking connection between del Pezzo surfaces and the exceptional $E$-series of root systems was uncovered by Manin in his seminal 1972 monograph \emph{Cubic Forms}. There, he identified these root systems within the Picard groups of del Pezzo surfaces, leveraging this structure to systematically investigate their geometry and arithmetic. Decades later, in 2001, the mathematical physicists Iqbal, Neitzke, and Vafa observed an analogous appearance of $E$-type root systems in the study of $\frac{1}{2}$-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 \emph{Mysterious Duality}.

Twenty years on, Hisham Sati and I discovered a similar root system pattern in the context of \emph{toroidifications} $\mathcal{T}^k S^4 := $ Map$(T^k, S^4)/ \! \! / T^k$ of the four-sphere $S^4$ for $0 \le k \le 8$, where $T^k = (S^1)^k$ is the $k$-torus. 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.

In this talk, I will present our findings on the root systems associated with toroidifications and describe an action of a maximal parabolic subgroup of a Lie group $G_k$ of type $E_k$ via rational self-equivalences of $\mathcal{T}^k S^4$. This action echoes Serganova and Skorobogatov’s work linking del Pezzo surfaces to homogeneous spaces $G_k/P_k$.