Four-Dimensional N=1 Gauge Dynamics

Four-dimensional $\mathcal N=1$ gauge dynamics is unusually rigid and unusually easy to overstate. Holomorphy, anomalies, zero-mode counting, and supersymmetric vacuum equations determine exact superpotentials and moduli constraints, but they do not by themselves determine a Kähler metric, a mass spectrum, or a duality. This chapter builds one controlled inference at a time, beginning with a complete SQCD theory card and ending with product groups and compactified semiclassics.

Enter this chapter

Begin with SQCD Fields, Symmetries, Global Data, and Classical Moduli to specify SQCD matter, symmetries, and classical vacuum coordinates.

Helpful background. ‘t Hooft Anomaly Matching; Global Form, Matter Representations, and the Faithful Gauge Group; Fermion Zero Modes, Index Data, and Selection Rules provide the anomaly matching, global gauge groups, and instanton selection rules used here.

Topics

1. SQCD Fields, Symmetries, Global Data, and Classical Moduli — Defines the theory before its infrared regime.
2. Pure Super-Yang–Mills Vacua and Domain Walls — Counts pure-gauge vacua and distinguishes wall questions.
3. Gaugino Condensation and Discrete Vacua — Tracks condensate phases and scale conventions.
4. Gauge Instantons, Fermion Zero Modes, and Condensates — Checks which correlators saturate instanton zero modes.
5. The Affleck–Dine–Seiberg Superpotential — Derives the runaway superpotential below the color threshold.
6. Quantum-Modified Moduli and s-Confinement — Separates deformed vacuum constraints from confinement with composites.
7. Infrared Phases and the Conformal Window — Organizes the flavor-dependent infrared regimes.
8. Chiral Gauge Theories and Anomaly Constraints — Uses anomalies without assuming a vectorlike meson description.
9. Dynamical Supersymmetry Breaking and Calculability — Distinguishes exact constraints from calculable breaking dynamics.
10. Product-Group and Quiver Gauge Dynamics — Tracks interactions and constraints shared between gauge nodes.
11. Compactification, Semiclassical Continuity, and Order of Limits — Tests which small-circle conclusions survive changes of limits.

How the ideas fit together

Holomorphic decoupling links apparently different SQCD regimes. Giving one flavor a supersymmetric mass changes the low-energy scale and reduces the flavor number, so proposed superpotentials or moduli constraints must reproduce the neighboring theory. This is a dynamical consistency test beyond matching symmetry charges. Even when it fixes an exact superpotential, it does not fix the Kähler metric, and therefore need not determine masses or the stability of an unprotected local minimum. Intriligator and Seiberg 1996, §§ 3–5.

Where to go next

Continue to Field-Theory Duality: Dictionaries, Operations, and Global Data to define both theories and the observables the proposed map should preserve.

References

• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric-Magnetic Duality.” Nuclear Physics B - Proceedings Supplements 45BC (1996): 1–28. doi:10.1016/0920-5632(95)00626-5. Open PDF.