Mysterious triality and the exceptional symmetry of loop spaces

In previous work (Sati and Voronov in Commun Math Phys 400:1915–1960, 2023. https://doi.org/10.1007/s00220-023-04643-7, in Adv Theor Math Phys 28(8):2491–2601, 2024. https://doi.org/10.4310/atmp.241119034750), we introduced Mysterious Triality, extending the Mysterious Duality (Iqbal et al. in Adv Theor Math Phys 5:769–808, 2002. https://doi.org/10.4310/ATMP.2001.v5.n4.a5) between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere \(S^4\), capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus \(T^k\), \(k \ge 1\), with its dynamics described via the iterated cyclic loop space \({\mathcal {L}}_c^k S^4\) of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type \(E_k\). In this paper, we discover much richer symmetry by extending the action of the Cartan subalgebra by symmetries of the equations of motion of \((11-k)\)d supergravity to a maximal parabolic subalgebra \(\mathfrak {p}_k^{k(k)}\) of the Lie algebra \(\mathfrak {e}_{k(k)}\) of the U-duality group. We do this by constructing the action on the rational homotopy model of the slightly more symmetric than \({\mathcal {L}}_c^k S^4\) toroidification \({\mathcal {T}}^k S^4\), which is another bookkeeping device for the equations of motion. To justify these results, we identify the minimal model of the toroidification \({\mathcal {T}}^k S^4\), generalizing the results of Vigué-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.