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# Mysterious Triality and Rational Homotopy Theory

Nov 2021

Mysterious Duality was discovered by Iqbal, Neitzke, and Vafa in 2001 as aconvincing, yet mysterious correspondence between certain symmetry patterns intoroidal compactifications of M-theory and del Pezzo surfaces, both governed bythe root system series $E_k$. It turns out that the sequence of del Pezzo surfaces is not the only sequenceof 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 homotopytheory. For this sequence of spaces, the correspondence between its $E_k$symmetry pattern and that of toroidal compactifications of M-theory is nolonger a mystery, as each space $\mathcal{L}_c^k S^4$ is naturally related tothe compactification of M-theory on the $k$-torus via identification of theequations of motion of $(11-k)$-dimensional supergravity as the definingequations of the Sullivan minimal model of $\mathcal{L}_c^k S^4$. This gives anexplicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa's Mysterious Duality between algebraicgeometry and physics into a triality, also involving algebraic topology. Viathis triality, duality between physics and mathematics is demystified, and themystery is transferred to the mathematical realm as duality between algebraicgeometry and algebraic topology. Now the question is: Is there an explicitrelation between the del Pezzo surfaces $\mathbb{B}_k$ and iterated cyclic loopspaces of $S^4$ which would explain the common $E_k$ symmetry pattern?

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