Mysterious Triality and Rational Homotopy Theory

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa (Adv Theor Math Phys 5:769–808, 2002) as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series E_k. It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same E_k symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere S⁴, and then forming its iterated cyclic loop spaces LckS4, within which we discover the E_k symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its E_k symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space LckS4 is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of (11 - k) -dimensional supergravity as the defining equations of the Sullivan minimal model of LckS4. This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa’s Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology.