Mysterious Triality and Rational Homotopy Theory

Abstract

Mysterious Duality was discovered by Iqbal, Neitzke, and Vafa in 2001 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 $S^4$, and then forming its iterated cyclic loop spaces $\mathcal{L}_c^k S^4$, 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 $\mathcal{L}_c^k S^4$ 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 $\mathcal{L}_c^k S^4$. 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. Now the question is: Is there an explicit relation between the del Pezzo surfaces $\mathbb{B}_k$ and iterated cyclic loop spaces of $S^4$ which would explain the common $E_k$ symmetry pattern?