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An Unorthodox Introduction to Supersymmetric Gauge Theory

Matthew J. Strassler

TL;DR

This set of notes offers a compact, RG-centric tour through three- and four-dimensional supersymmetric gauge theories, beginning with classical Wess–Zumino and XYZ models and building up to abelian and non-abelian gauge theories. It emphasizes holomorphy, nonrenormalization, and the Wilsonian renormalization framework to explain fixed points, dualities, and moduli spaces, with explicit treatment of 3d and 4d phenomena such as solitons, vortex confinement, and NSVZ-type beta functions. The author highlights exact results, infrared dualities (e.g., XYZ vs. SQED in 3d), and the intricate web of fixed points and marginal operators that govern the IR physics, while noting several topics left for future exploration. Overall, the notes illuminate how holomorphy and RG ideas yield powerful constraints and dualities that shape the dynamics of supersymmetric gauge theories with broad implications for quantum field theory and string theory.

Abstract

Numerous topics in three and four dimensional supersymmetric gauge theories are covered. The organizing principle in this presentation is scaling (Wilsonian renormalization group flow.) A brief introduction to scaling and to supersymmetric field theory, with examples, is followed by discussions of nonrenormalization theorems and beta functions. Abelian gauge theories are discussed in some detail, with special focus on three-dimensional versions of supersymmetric QED, which exhibit solitons, dimensional antitransmutation, duality, and other interesting phenomena. Many of the same features are seen in four-dimensional non-abelian gauge theories, which are discussed in the final sections. These notes are based on lectures given at TASI 2001.

An Unorthodox Introduction to Supersymmetric Gauge Theory

TL;DR

This set of notes offers a compact, RG-centric tour through three- and four-dimensional supersymmetric gauge theories, beginning with classical Wess–Zumino and XYZ models and building up to abelian and non-abelian gauge theories. It emphasizes holomorphy, nonrenormalization, and the Wilsonian renormalization framework to explain fixed points, dualities, and moduli spaces, with explicit treatment of 3d and 4d phenomena such as solitons, vortex confinement, and NSVZ-type beta functions. The author highlights exact results, infrared dualities (e.g., XYZ vs. SQED in 3d), and the intricate web of fixed points and marginal operators that govern the IR physics, while noting several topics left for future exploration. Overall, the notes illuminate how holomorphy and RG ideas yield powerful constraints and dualities that shape the dynamics of supersymmetric gauge theories with broad implications for quantum field theory and string theory.

Abstract

Numerous topics in three and four dimensional supersymmetric gauge theories are covered. The organizing principle in this presentation is scaling (Wilsonian renormalization group flow.) A brief introduction to scaling and to supersymmetric field theory, with examples, is followed by discussions of nonrenormalization theorems and beta functions. Abelian gauge theories are discussed in some detail, with special focus on three-dimensional versions of supersymmetric QED, which exhibit solitons, dimensional antitransmutation, duality, and other interesting phenomena. Many of the same features are seen in four-dimensional non-abelian gauge theories, which are discussed in the final sections. These notes are based on lectures given at TASI 2001.

Paper Structure

This paper contains 32 sections, 124 equations, 44 figures.

Table of Contents

  1. Introduction
  2. Classical theory
  3. Free massless fields
  4. Free massive fields
  5. Supersymmetry! The Wess-Zumino model
  6. The XYZ model
  7. XYZ with a mass
  8. Solitons in XYZ
  9. A more general form
  10. Perturbation Theory
  11. The quantum Wess-Zumino model
  12. Wess-Zumino model in $d=3$
  13. Dimensions and R-charge
  14. The quantum XYZ model in $d=3$
  15. Abelian gauge theories
  16. ...and 17 more sections

Figures (44)

  • Figure 1: The effect of a mass term grows in the infrared.
  • Figure 2: The scaling flow of two masses, with $\rho = m_1/m_2=\nu_1/\nu_2$.
  • Figure 3: The moduli space of the XYZ model.
  • Figure 4: The moduli space once a mass for $X$ is added.
  • Figure 5: The flow at the origin of moduli space; the effect of the $Y^2Z^2$ superpotential renormalizes away in the far infrared.
  • ...and 39 more figures