On Hexagonal Structures in Higher Dimensional Theories

TL;DR

This paper explores the geometric structures, especially hexagonal and octonionic, underlying various Lie groups in higher-dimensional theories relevant to particle physics, highlighting their roles in compactification and gauge symmetries.

Contribution

It uncovers the connection between hexagonal structures, octonions, and key Lie groups in high-energy physics, providing a geometric perspective on symmetry origins.

Findings

Identification of hexagonal structures in Lie groups like SU(2), SU(3), G_2, Spin(7), SO(8), E_8, SO(32)

Relation between these structures and octonion division algebra

Implication of these structures in compactification and gauge symmetry interpretation

Abstract

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2),SU(3), G_2, Spin(7), SO(8) as well as E_8 and SO(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.