Z_N-Invariant Subgroups of Semi-Simple Lie Groups

TL;DR

This paper uses computational methods to identify $Z_N$-invariant subgroups of $E_8$, relevant for M-theory compactifications, revealing potential gauge groups similar to the Standard Model.

Contribution

It introduces a Mathematica-based procedure to find $Z_N$-invariant subgroups of $E_8$, applicable to various $Z_N$ groups acting on root lattices, with specific examples for $Z_7$.

Findings

Identified $Z_7$-invariant subgroups of $E_8$ relevant for orbifold compactification.

Developed a general method for $Z_N$-invariant subgroup identification.

Demonstrated applicability to multiple Lie group factors.

Abstract

We employ Mathematica to find $Z_N$-invariant subgroups of $E_8$ for application in M-theory. These $Z_N$-invariant subgroups are phenomenologically important and in some cases they resemble the gauge groups of our real world. We present a specific example of $Z_7$-invariant subgroups of $E_8$, which turn up in orbifold compactification of M-theory. Moreover, the procedure can be applied for any $Z_N$ group that acts by shifts (translations) in the root lattice of semisimple Lie groups with $A_n,B_n,C_n,D_n,E_6,E_7$ and $E_8$ factors.