Gauge Theory: Instantons, Monopoles, and Moduli Spaces

TL;DR

This paper provides a comprehensive mathematical overview of gauge theory focusing on instantons and monopoles, connecting physical concepts with advanced mathematical structures like moduli spaces and Seiberg-Witten equations.

Contribution

It offers a clear, self-contained exposition of key gauge theory topics, emphasizing the mathematical structures underlying instantons and monopoles with connections to physics.

Findings

Mathematical structure of Seiberg-Witten monopole equations explained

Donaldson's theorem on monopole moduli spaces discussed

Compactification of ASD connection spaces analyzed

Abstract

In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematics. We give a clear and essentially self-contained exposition of the mathematical structure of the Seiberg-Witten monopole equations. Other topics include Donaldson's theorem on moduli spaces of monopoles, compactification of spaces of ASD connections, The Abelian monopole equations, and Abelian Higgs vortices.