The Hidden M-Group

TL;DR

This paper constructs a super-Lie group integrating the hidden M-algebra related to 11D supergravity, enabling modeling of super-exceptional spacetimes and exploring their compactifications, with rigorous mathematical and computational foundations.

Contribution

It explicitly integrates the hidden M-algebra into a super-Lie group and develops a rigorous, physics-inspired framework for super-exceptional geometry and spacetime compactifications.

Findings

Construction of the super-Lie group for the hidden M-algebra.

Explicit computer-checked derivation of the M-theory 3-form.

Description of lattice subgroups enabling toroidal compactification.

Abstract

Following arguments that the (hidden) M-algebra serves as the maximal super-exceptional tangent space for 11D supergravity, we make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of the ''decomposed'' M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries. As a simple but consequential application, we highlight how to describe lattice subgroups $\mathbb{Z}^{k \leq 528}$ of the hidden M-group that allow to toroidially compactify also the ''hidden'' dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality. In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the ''decompose'' M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice. Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations is the dimensional reduction of the hidden M-algebra to a ''hidden IIA-algebra'' which in a companion article we explain as the exceptional extension of the T-duality doubled super-spacetime.