Schreiber The Hidden M-Group

An article that we are finalizing at CQTS:

---

• Grigorios Giotopoulos, $\;$ Hisham Sati $\;$ and $\,$ Urs Schreiber:

  
  

  The Hidden M-Group

  
  

  download:

  • main: pdf $\;$ (manuscript)

  • anc: nb $\;$ (Mathematica notebook)

  • arXiv:2411.11963

---

Abstract. Following arguments that the M-algebra – both in its basic version and via its “hidden” extension – serves as the maximal super-exceptional tangent space for 11D supergravity, we here make it explicit as a (super-Lie) group 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 which allow to toroidially compactify also the “hidden” dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality.

To deal with subtleties in the construction we also (1.) provide a computer-checked re-derivation of the “decomposed” 3-form, and we (2.) present a streamlined conception of super-Lie groups which is both rigorous while still close to physics practice.

Thereby this article is mostly a combined review of some 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 an exceptional extension of the T-duality doubled super-spacetime.

Followup to:

• The M-algebra completes the hierarchy of Super-Exceptional Tangent Spaces

• Super-Lie-infinity T-Duality and M-Theory

• Super-exceptional embedding construction of the M5-brane

• Super-exceptional M5-brane model – Emergence of SU(2)-flavor sector