On Maximal, Universal and Complete Extensions of Yang-Mills-Type Theories

TL;DR

This paper advances the classification of Yang-Mills-type theory extensions by establishing categorical characterizations and analyzing properties like maximality and universality, with implications for the structure of extensions in particle physics.

Contribution

It introduces categorical criteria for classifying extensions of Yang-Mills theories and proves density and maximality results, including independence from trivial extensions under the Axiom of Choice.

Findings

Maximality and universality are dense properties in extension classes.

Existence and obstruction problems are fully solved for almost coherent extensions.

Maximality theorems are established, some relying on the Axiom of Choice.

Abstract

In this paper we continue the program on the classification of extensions of the Standard Model of Particle Physics started in arXiv:2007.01660. We propose four complementary questions to be considered when trying to classify any class of extensions of a fixed Yang-Mills-type theory $S^G$: existence problem, obstruction problem, maximality problem and universality problem. We prove that all these problems admits a purely categorical characterization internal to the category of extensions of $S^G$. Using this we show that maximality and universality are dense properties, meaning that if they are not satisfied in a class $\mathcal{E}(S^G;\hat{G})$, then they are in their "one-point compactification" $\mathcal{E}(S^G;\hat{G})\cup \hat{S}$ by a specific trivial extension $\hat{S}$. We prove that, by means of assuming the Axiom of Choice, one can get another maximality theorem, now independent of the trivial extension $\hat{S}$. We consider the class of almost coherent extensions, i.e, complete, injective and of pullback-type, and we show that for it the existence and obstruction problems have a complete solution. Using again the Axiom of Choice, we prove that this class of extensions satisfies the hypothesis of the second maximality theorem.