Geometry of classical Higgs fields

TL;DR

This paper explores the geometric structure of classical Higgs fields in gauge theory, focusing on symmetry breaking via principal bundle reductions and the associated covariant differential.

Contribution

It provides a detailed geometric analysis of Higgs fields as reductions of principal bundles and examines the covariant differential in this context.

Findings

Higgs fields correspond to reductions of principal bundles.

Existence of a global section of P/H signifies symmetry breaking.

The paper clarifies the geometric role of Higgs fields in gauge theories.

Abstract

In gauge theory, Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P, a symmetry breaking is defined as the reduction of a structure group of this principal bundle to a subgroup H of exact symmetries. This reduction takes place iff there exists a global section of the quotient bundle P/H. It is a classical Higgs field. A metric gravitational field exemplifies such a Higgs field. We summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields. Our goal is the particular covariant differential in the presence of a Higgs field.