Exceptional Structures in Mathematics and Physics and the Role of the Octonions

TL;DR

This paper explores the role of octonions in exceptional mathematical structures and their potential connection to a unified Theory of Everything, focusing on octonionic spinors and supersymmetry realizations.

Contribution

It systematically investigates octonionic properties and their implications for supersymmetry, particularly defining an octonionic M-algebra with unique properties.

Findings

Octonionic M-algebra can be consistently defined with real and octonionic structures.

Octonionic p-form identities induce striking properties in the M-algebra.

Octonions underpin exceptional structures potentially linked to a Theory of Everything.

Abstract

There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate "exceptional" formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above. In this context it is quite motivating to systematically investigate the properties of octonionic spinors and the octonionic realizations of supersymmetry. In particular the $M$-algebra can be consistently defined for two structures only, a real structure, leading to the standard $M$-algebra, and an octonionic structure. The octonionic version of the $M$-algebra admits striking properties induced by octonionic $p$-forms identities.