Lectures on instantons

TL;DR

The notes provide a comprehensive, self-contained treatment of Yang–Mills and (super)Yang–Mills instantons in four dimensions, deriving explicit one-instanton solutions for $SU(2)$ and $SU(N)$, their zero modes, and the full moduli-space measure, including bosonic and fermionic sectors. They develop the one-loop determinant analysis and show how supersymmetry enforces cancellations leading to exact or scheme-dependent beta functions, with a vanishing beta function in ${\cal N}=4$ theory; they also illuminate the role of instantons in key phenomenological issues such as the $ heta$-angle in QCD, the $U(1)$ problem, and Higgs-induced large-instanton suppression. The lectures connect these nonperturbative constructs to tunnelling, false-vacuum decay, and phase transitions, and discuss extensions to constrained instantons and multi-instanton configurations via ADHM, as well as the ${\cal N}=4$ theory’s Euclidean incarnation and its implications for AdS/CFT. Overall, the work provides a rigorous toolkit for computing nonperturbative effects in gauge theories and exploring their physical consequences.

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 θ-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