Topological Concepts in Gauge Theories

TL;DR

This paper introduces topological concepts in gauge theories, discussing key topological objects like vortices, monopoles, and instantons, and explores their roles in phenomena such as confinement in Quantum Chromodynamics.

Contribution

It provides a comprehensive overview of topological methods in gauge theories, highlighting their formal construction and applications to confinement and other physical phenomena.

Findings

Analysis of topological objects in gauge theories

Discussion of topological methods in QCD confinement

Introduction to homotopy in physics

Abstract

In these lecture notes, an introduction to topological concepts and methods in studies of gauge field theories is presented. The three paradigms of topological objects, the Nielsen-Olesen vortex of the abelian Higgs model, the 't Hooft-Polyakov monopole of the non-abelian Higgs model and the instanton of Yang-Mills theory, are discussed. The common formal elements in their construction are emphasized and their different dynamical roles are exposed. The discussion of applications of topological methods to Quantum Chromodynamics focuses on confinement. An account is given of various attempts to relate this phenomenon to topological properties of Yang-Mills theory. The lecture notes also include an introduction to the underlying concept of homotopy with applications from various areas of physics.