Mysterious Triality and M-Theory

In a previous paper [168], 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 [103], 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 L^k _c S⁴, producing a series of data analogous to that given by the del Pezzo surfaces B_k, for k = 0,…, 8. With the mathematical constructions established in [168], in this companion physics paper we present novel connections to Mtheory 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.