Elements of Finite Order in Lie Groups and Discrete Gauge Symmetries

TL;DR

This paper explores how finite order elements in Lie groups relate to discrete gauge symmetries in supersymmetric grand unified theories, identifying specific elements that generate matter parities.

Contribution

It applies Kac's theory of finite order elements to discrete gauge symmetries in supersymmetric GUT models, linking mathematical structures to physical symmetry properties.

Findings

Identifies EFO generating matter parities in SO(10) and E6 models

Ensures discrete anomaly cancellation conditions are satisfied

Provides a mathematical framework for understanding discrete symmetries in GUTs

Abstract

We apply Kac's theory of elements of finite order (EFO) in Lie groups to the description of discrete gauge symmetries in various supersymmetric grand unified models. Taking into account the discrete anomaly cancellation conditions, we identify the EFO which generate certain matter parities in the context of the supersymmetric $SO(10)$ and $E_6$ models.