Mysterious Triality and M-Theory

In a previous paper, we introduced Mysterious Triality as an extension, via algebraic topology in the form of rational homotopy theory, of Mysterious Duality by Iqbal, Neitzke, and Vafa, which provides connections between physics, in the form of dimensional reduction of M-theory, and algebraic geometry, in the form of intersection theory on del Pezzo surfaces. The starting point for that connection to rational homotopy theory is the description of M-theory dynamics using the 4-sphere, via Hypothesis H. This progresses to dimensional reduction of M-theory on tori $T^k$ with its dynamics described via cyclic loop spaces of the 4-sphere $\mathcal{L}_c^k S^4$, producing a series of data analogous to that given by the del Pezzo surfaces $\mathbb{B}_k$, for $k=0, \dots, 8$. With the mathematical constructions established in the previous paper, in this companion physics paper we present novel connections to M-theory that enhance the triality, including those strengthening the duality. This uncovers interesting ties between algebraic geometry, algebraic topology, and M-theory and provides tantalizing links. We further expand on the extension of the duality and triality to the Kac-Moody setting.