Field-Theory Duality: Dictionaries, Operations, and Global Data
Dual descriptions can use different fields and couplings to encode the same physics, but the regime and matched objects must be specified. This chapter develops the comparison through operator maps, global data, gauging, interfaces, and deformations before applying it to named dual pairs.
Enter this chapter
Section titled “Enter this chapter”Begin with Duality Claims, Dictionaries, Regimes, and Evidence to define both theories and the observables the proposed map should preserve.
Helpful background. Symmetry, Gauge Redundancy, and Duality; Global Form, Matter Representations, and the Faithful Gauge Group; ‘t Hooft Anomaly Matching; Boundaries, Interfaces, and Domain Walls provide the symmetry versus redundancy, global groups, anomalies, and interfaces used here.
Topics
Section titled “Topics”- Duality Claims, Dictionaries, Regimes, and Evidence — States the objects and regime of a proposed equivalence.
- Exact, Infrared, and Emergent Equivalence Claims — Separates full equivalence from infrared or protected-sector claims.
- Abelian Electric–Magnetic Dualization, Charge Lattices, and Global Form — Tracks integral electric and magnetic charges.
- Duality Operations: Gauging, Quotients, and Orbifolds — Identifies the extra data needed when gauging or quotienting.
- Generalized Symmetries, Global Forms, and Anomalies under Duality — Matches genuine operators and generalized-symmetry anomalies.
- Duality Defects, Walls, Interfaces, and Fusion — Treats duality walls as coupled physical systems.
- Deformations, Compactification, and Duality Flows — Checks whether deformation and dualization lead to the same endpoint.
- Duality Checks, Evidence Independence, Status, and Failure Modes — Distinguishes complementary tests from logically repeated ones.
How the ideas fit together
Section titled “How the ideas fit together”Two theories can share a local gauge algebra and perturbative operators while differing in genuine line operators and allowed bundles. An operator dictionary restricted to local fields can therefore miss a physical distinction. Gauging a shared symmetry makes that omission especially consequential because the background response becomes dynamical. Global form, anomaly, and contact-term information must travel through the comparison, and matching one protected quantity remains a test of that quantity rather than proof of every observable. Aharony, Seiberg, and Tachikawa 2013, §§ 1–2.
Where to go next
Section titled “Where to go next”Continue to N=1 Duality, RG Fixed Points, and Protected SCFT Data to write the electric and magnetic SQCD theories in compatible conventions.