Symmetric Spaces of Exceptional Groups

TL;DR

This paper investigates the origins of 12 symmetric spaces associated with exceptional Lie groups, linking their existence to octonionic structures, Freudenthal's magic square, and supergravity theories across various spacetime dimensions.

Contribution

It provides a unified explanation for the existence of these symmetric spaces using algebraic and physical frameworks, connecting exceptional groups to octonions and supergravity.

Findings

12 symmetric spaces linked to exceptional groups explained

Octonionic structures account for G2 and F4 cases

Supergravity theories relate to E6, E7, E8 symmetric spaces

Abstract

We adress the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1+2 cases for $G_2$ and $F_4$ respectively are easily explained from the octonionic nature of these groups. The 4+3+2 cases on the $E_{6,7,8}$ series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in $5, 4$ and 3 spacetime dimensions.