Holomorphy, Anomalies, and Exact Quantum Constraints
In four-dimensional theories, holomorphy can turn symmetry and anomaly data into exact quantum information—but only after the object being constrained has been named precisely. A Wilsonian superpotential, a 1PI vertex, a holomorphic gauge coupling, and a canonically normalized coupling obey different statements. This chapter develops a reliable workflow for deciding what is exact, what remains an undetermined constant or branch choice, and what requires independent dynamical input.
Enter this chapter
Section titled “Enter this chapter”Begin with Holomorphic Couplings and Background Superfields to promote couplings to background chiral fields before applying holomorphy.
Helpful background. Regulated Jacobians and Measure Variation; Beta Functions, Running Masses, and Field Anomalous Dimensions; Fermion Zero Modes, Index Data, and Selection Rules provide the anomalous measure transformations, RG functions, and fermion zero modes used here.
Topics
Section titled “Topics”- Holomorphic Couplings and Background Superfields — Uses symmetry charges of couplings as constraints.
- Nonrenormalization Theorems: Wilsonian, 1PI, and Infrared Scope — States which local F-terms are protected and under what infrared assumptions.
- R-Symmetry, Anomalies, and the Holomorphic Scale — Tracks anomalous symmetries through the complex scale.
- Holomorphic and Canonical Couplings and the NSVZ Relation — Distinguishes holomorphic and canonically normalized running.
- Instanton Zero Modes and Selection Rules for Exact Terms — Tests whether an instanton can generate a superpotential term.
- Nonperturbative Superpotentials and Exactness Conditions — Combines holomorphy with asymptotics and remaining constants.
- Holomorphic Decoupling, Integrating In and Out, and Scale Matching — Matches exact descriptions across supersymmetric thresholds.
- Quantum Chiral Rings and Konishi-Type Anomaly Constraints — Derives quantum relations from anomalous chiral transformations.
How the ideas fit together
Section titled “How the ideas fit together”A holomorphic constraint applies to a specified Wilsonian object. Canonically normalizing fields introduces wavefunction factors, so a protected holomorphic superpotential does not imply constant physical couplings. Infrared effects can also distinguish a 1PI vertex from the local Wilsonian action. Symmetry and holomorphy restrict possible nonperturbative terms, but their coefficients, branches, or existence may need decoupling limits and zero-mode tests. Exactness therefore belongs to the stated object and assumptions, rather than to every quantity bearing the same coupling name. Seiberg 1993, §§ 2–3.
Where to go next
Section titled “Where to go next”Continue to Supersymmetry Breaking and Controlled Deformations to relate nonzero auxiliary fields to vacuum energy and the goldstino.