Topology of Fibre bundles and Global Aspects of Gauge Theories

TL;DR

This paper introduces the mathematical framework of fibre bundles and topology to understand global features of gauge theories like monopoles and instantons, aimed at beginning PhD students.

Contribution

It provides an accessible introduction to fibre bundles, connections, topology, and their application to gauge theory phenomena such as monopoles and instantons.

Findings

Application of topology to classify monopoles and instantons

Introduction of characteristic classes and index theorems in gauge theories

Educational resource for beginning PhD students in mathematical physics

Abstract

In these lecture notes we will try to give an introduction to the use of the mathematics of fibre bundles in the understanding of some global aspects of gauge theories, such as monopoles and instantons. They are primarily aimed at beginning PhD students. First, we will briefly review the concept of a fibre bundle and define the notion of a connection and its curvature on a principal bundle. Then we will introduce some ideas from topology such as homotopy, topological degree and characteristic classes. Finally, we will apply these notions to the bundle setup corresponding to monopoles and instantons. We will end with some remarks on index theorems and their applications and some hints towards a bigger picture.