  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.
