Lectures on instantons

TL;DR

This paper provides comprehensive lecture notes on instantons in (super) Yang-Mills theories, covering their solutions, zero modes, measures, and applications to QCD and quantum mechanics, serving as an educational resource.

Contribution

It offers a self-contained, detailed derivation of instanton solutions and their applications in gauge theories, extending previous reviews with updated insights.

Findings

Derivation of regular and singular instanton solutions for SU(2) and SU(N)

Analysis of zero modes and instanton measures in supersymmetric theories

Applications to -angle, U(1) problem, and tunneling phenomena in gauge theories

Abstract

This is a self-contained set of lecture notes on instantons in (super) Yang-Mills theory in four dimensions and in quantum mechanics. First the basics are derived from scratch: the regular and singular one-instanton solutions for Yang-Mills theories with gauge groups SU(2) and SU(N), their bosonic and fermionic zero modes, the path integral instanton measure, and supersymmetric Yang-Mills theories in Euclidean space. Then we discuss applications: the \theta-angle of QCD, the solution of the U(1) problem, the way Higgs fields solve the large-instanton problem, and tunneling and phase transitions in quantum mechanics and in nonabelian gauge theories. These lecture notes are an extension of a review on Yang-Mills and D-instantons written in 2000 by both authors and A.Belitsky