Mysterious Triality and the Exceptional Symmetry of Loop Spaces

TL;DR

This paper explores the extended symmetry structures in rational homotopy theory related to M-theory, revealing actions of parabolic subalgebras of exceptional Lie algebras on toroidified spheres, which model supergravity equations.

Contribution

It extends the algebraic symmetry framework of M-theory by incorporating parabolic subalgebras acting on toroidified spheres in rational homotopy theory.

Findings

Identified the minimal model of the toroidification of S^4.

Established an algebraic toroidification/totalization adjunction.

Discovered an action of parabolic subalgebras of E_k on toroidified S^4.

Abstract

In previous work, we introduced Mysterious Triality, extending the Mysterious Duality of Iqbal, Neitzke, and Vafa 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 data of the Cartan subalgebra to a maximal parabolic subalgebra $\mathfrak{p}_k^{k(k)}$ of the split real form $\mathfrak{e}_{k(k)}$ of the exceptional Lie algebra of type $E_k$ by exhibiting an action, in rational homotopy category, of $\mathfrak{p}_k^{k(k)}$ on the slightly more symmetric than $\mathcal{L}_c^k S^4$ toroidification $\mathcal{T}^k S^4$. This action universally represents symmetries of the equations of motion of supergravity in the reduction of M-theory to $11-k$ dimensions. Along the way, we identify the minimal model of the toroidification $\mathcal{T}^k S^4$, generalizing the results of Vigu\'{e}-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.