Constraints on Unified Gauge Theories from Noncommutative Geometry

TL;DR

This paper explores how noncommutative geometry constrains the extension of gauge theories, indicating that larger unified gauge groups are incompatible unless the fermionic content is expanded.

Contribution

It demonstrates that noncommutative geometry imposes strict limitations on extending the Standard Model to larger gauge groups without increasing fermionic degrees of freedom.

Findings

Larger gauge groups are incompatible with noncommutative geometry constraints.

The Higgs field emerges naturally as a gauge boson in this framework.

Extensions require enlarging the fermionic sector.

Abstract

The Connes and Lott reformulation of the strong and electroweak model represents a promising application of noncommutative geometry. In this scheme the Higgs field naturally appears in the theory as a particular `gauge boson', connected to the discrete internal space, and its quartic potential, fixed by the model, is not vanishing only when more than one fermion generation is present. Moreover, the exact hypercharge assignments and relations among the masses of particles have been obtained. This paper analyzes the possibility of extensions of this model to larger unified gauge groups. Noncommutative geometry imposes very stringent constraints on the possible theories, and remarkably, the analysis seems to suggest that no larger gauge groups are compatible with the noncommutative structure, unless one enlarges the fermionic degrees of freedom, namely the number of particles.