Duality in Supersymmetric Yang-Mills Theory

TL;DR

Peskin surveys the role of effective Lagrangians in strongly coupled supersymmetric gauge theories, drawing connections between holomorphy, vacuum structure, and nonperturbative dynamics. The Seiberg-Witten solution for N=2 SU(2) YM reveals how duality and monodromy encode the Coulomb phase via a family of elliptic curves; this geometric framework is extended to theories with matter and larger gauge groups. The talk then develops non-Abelian electric-magnetic duality (Seiberg duality) in N=1 SQCD, including holomorphic decoupling and anomaly matching, and discusses fixed points, conformal windows, and generalizations to SO/Sp groups and chiral matter. Together, the material demonstrates a unifying picture where strong coupling phenomena, dual descriptions, and moduli-space geometry illuminate the infrared behavior of supersymmetric gauge theories with wide-ranging implications for quantum field theory and string theory.

Abstract

These lectures provide an introduction to the behavior of strongly-coupled supersymmetric gauge theories. After a discussion of the effective Lagrangian in nonsupersymmetric and supersymmetric field theories, I analyze the qualitative behavior of the simplest illustrative models. These include supersymmetric QCD for $N_f < N_c$, in which the superpotential is generated nonperturbatively, N=2 SU(2) Yang-Mills theory (the Seiberg-Witten model), in which the nonperturbative behavior of the effective coupling is described geometrically, and supersymmetric QCD for N_f large, in which the theory illustrates a non-Abelian generalization of electric-magnetic duality. [Lectures presented at the 1996 TASI Summer School, to appear in the proceedings.]

Figures (12)

• Figure 1: Form of the potential for supersymmetric Yang-Mills theory with $N_f< N_c$.
• Figure 2: Determination of the effective coupling constant in the weak-coupling limit of the Seiberg-Witten model.
• Figure 3: (a) Spectrum of $W$ bosons, monopoles, and dyons in the weak-coupling limit of the Seiberg-Witten model with $\theta = 0$. (b) Transformation of the spectrum of monopole and dyon states as we turn on $\theta$ or rotate $u$ in the weak-coupling region.
• Figure 4: Dependence of monopole and dyon masses on $u$ along the positive real axis of the $u$ plane.
• Figure 5: Relation of the monodromies about the three singularities of $\tau(u)$.