The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking

TL;DR

This paper provides an accessible introduction to gauge theory, spontaneous symmetry breaking, and the Higgs mechanism, tailored for mathematicians with basic knowledge of Lie groups and manifolds.

Contribution

It offers a clear mathematical exposition of the Higgs mechanism and symmetry breaking, making these physics concepts more accessible to mathematicians.

Findings

Clarifies the Higgs mechanism in the context of gauge theory

Explains symmetry breaking for arbitrary compact gauge groups

Details the electroweak interaction case with G=SU(2)xU(1)

Abstract

These notes form part of a lecture course on gauge theory. The material covered is standard in the physics literature, but perhaps less well-known to mathematicians. The purpose of these notes is to make spontaneous symmetry breaking and the Higgs mechanism of mass generation for elementary particles more easily accessible to mathematicians interested in theoretical physics. We treat the general case with an arbitrary compact gauge group G and an arbitrary number of Higgs bosons and explain the situation in the classic case of the electroweak interaction where G=SU(2)xU(1). Prerequisites are only a basic knowledge of Lie groups and manifolds. No prior knowledge of gauge theory or bundle theory is assumed.