Integer versions of Yang-Mills theory

TL;DR

This paper explores an integer-based version of Yang-Mills theory, focusing on the classification and representation theory of gauge groups relevant to the standard model, aiming to address 't Hooft's question about the universe's fundamental structure.

Contribution

The paper provides a detailed classification and analysis of integer gauge groups for Yang-Mills theory, connecting group theory with physical implications at the Planck scale.

Findings

Classification of integer gauge groups relevant to Yang-Mills theory

Analysis of the representation theory of these groups

Insights into the physical implications at the Planck scale

Abstract

In a recent paper, 't Hooft asks for an integer version of Yang-Mills theory, in the belief that this is the way the universe really is at the Planck scale. Specifically, he asks for an integer version of the gauge group of the standard model. Such groups were completely classified more than 100 years ago, and here I detail the group theory and representation theory (and some of the physics) of the specific case that I believe answers 't Hooft's question.