Instantons Versus Supersymmetry: Fifteen Years Later

TL;DR

The paper reviews how instanton calculus in supersymmetric gauge theories, reinforced by holomorphy and anomaly considerations, yields exact nonperturbative results across weak and strong coupling and clarifies mechanisms for dynamical supersymmetry breaking. It systematically develops the superinstanton formalism, including moduli, collective coordinates, and the exact instanton measure, and applies these tools to compute the NSVZ beta function, gluino condensates, and instanton-induced superpotentials in various SQCD-like models. It then surveys a spectrum of calculable weak- and strong-coupling SUSY-breaking scenarios (e.g., 3-2, SU(5) with two generations, 4-1, and ISS), emphasizing how nonperturbative dynamics lift classical flat directions and stabilize or destabilize vacua. The discussion highlights the central role of domain walls, central charges, and anomalies in shaping the vacuum structure and nonperturbative phenomena, and it outlines the boundaries and open questions in fully understanding SUSY breaking in strongly coupled regimes.

Abstract

An introduction to the instanton formalism in supersymmetric gauge theories is given. We explain how the instanton calculations, in conjunction with analyticity in chiral parameters and other general properties following from supersymmetry, allow one to establish exact results in the weak and strong coupling regimes. Some key applications are reviewed, the main emphasis is put on the mechanisms of the dynamical breaking of supersymmetry.

Figures (5)

• Figure 1: A typical two-loop supergraph. The solid lines denote the propagators of the quantum superfields in the (anti)instanton background. We rely only on the most general features of the supersymmetric background field technique. For a pedagogical introduction to supergraphs and supersymmetric background field technique the reader is referred to Ref. SSOT.
• Figure 2: One-instanton contribution in the SU(2) model with one flavor.
• Figure 3: One-instanton contribution in the ${\cal N}=2$ theory (with no matter hypermultiplets).
• Figure 4: The 3-2 model: the zero mode structure in the field of the anti-instanton in the gauge groups SU(3) and SU(2), respectively. $\lambda$ denotes the SU(3) gluinos in the first case and SU(2) gluinos in the second.
• Figure 5: The instanton saturation of the correlation function (\ref{['mfcf']}) in the one-generation SU(5) model.