Field-Theory Duality: Dictionaries, Operations, and Global Data

A field-theory duality is a claim that two apparently different descriptions encode the same specified physics in a specified regime. The claim is meaningful only after both theories, the observable dictionary, global data, parameter map, limits, and evidence are stated. This chapter supplies that reusable framework before later chapters examine Seiberg duality, electric–magnetic duality, mirror symmetry, or duality webs. Four-dimensional $\mathcal N=1$ SQCD supplies the canonical infrared example: the preceding chapter’s phase analysis locates the conformal window, while Seiberg 1995, §§2 and 3.2 (PDF) constructs the electric and magnetic descriptions and tests their mass-deformation map.

Helpful background. Review symmetry, redundancy, and duality, global form, ’t Hooft anomaly matching, and boundaries and interfaces.

Enter through the claim that is actually being made

Before comparing calculations, classify the statement along the relevant axes. The rows below overlap: for example, an infrared duality may also be emergent, and either claim may be tested only in a protected subsector.

Claim	Meaning	What it does not imply
Exact equivalence	The declared QFTs agree, including the specified global and extended sectors and any allowed invertible factors or counterterms.	That the descriptions use the same elementary fields or weak-coupling expansion.
Infrared duality	Two ultraviolet theories flow to the same infrared theory after named decoupled sectors are included.	Equality of ultraviolet correlators or irrelevant couplings.
Emergent equivalence	A common description appears only below a scale or at a fixed point.	A microscopic change of variables.
Reformulation	Two presentations are related by an exact transformation, often with a changed polarization.	A new dynamical conjecture.
Protected-subsector match	A cohomology, index, topological twist, or BPS sector agrees.	Equality of the full unprotected theory.
Gauging or quotient relation	One theory is obtained from another by summing over backgrounds or projecting sectors.	Invertible equivalence; information may be lost or new twisted sectors added.

The equivalence taxonomy sharpens these distinctions. If the scope cannot be named, the proposed duality is not yet a testable statement.

The minimum complete comparison

The claims and dictionaries page builds a comparison from eight coupled blocks:

1. Theory definitions: dimension, signature, field content, action or fixed-point data, global gauge group, discrete choices, and allowed backgrounds.
2. Regime: exact, infrared, large-parameter, compactified, topologically twisted, or restricted to a protected sector.
3. Parameters: masses, couplings, theta angles, FI terms, scales, and counterterms, including complex conjugations and additive contact terms.
4. Local operators: quantum numbers, normalization, chiral-ring relations, descendants, and accidental or decoupled operators.
5. Extended operators: genuine lines, surfaces, defects, endpoints, fusion, and screening.
6. Symmetries and anomalies: ordinary, higher-form, higher-group, and noninvertible structures in background fields.
7. Vacua and states: moduli branches, BPS charges, superselection sectors, and boundary conditions.
8. Evidence and falsifiers: independent checks, shared assumptions, unmatched sectors, and a condition that would disprove the proposed scope.

A blank block is not neutral. It either narrows the claim or marks unfinished work.

Choose a route through the chapter

I need to formulate a new duality. Begin with claims, dictionaries, regimes, and evidence, then classify its exact or infrared scope. Finish with checks and failure modes.

I need to compare electric and magnetic descriptions. Use abelian dualization, charge lattices, and global form, then generalized symmetries and anomalies. A Lie-algebra map without a genuine-line map is incomplete.

I am gauging a common symmetry. Use gauging, quotients, and orbifolds. Track twisted sectors, residual symmetries, topological sectors, and anomaly counterterms on both sides.

I am deforming or compactifying a known pair. Use deformations, compactification, and flows. Reaching equivalent endpoints is necessary but not sufficient: the induced dictionaries along the two paths must agree through a stated composition map after scale hierarchies, vacuum choices, accidental sectors, and noncommuting limits are included.

I want a geometric realization of the map. Use duality walls and interfaces. Transporting operators through an invertible wall can realize a duality, while noninvertible fusion can instead produce a sum or projector.

Three layers of consistency

The strongest comparisons test different physical layers.

Local layer

Match conserved currents, relevant and protected operators, chiral rings, OPE coefficients where available, moduli spaces, and RG data. Anomaly matching is necessary for an infrared duality with the same unbroken global symmetry, but it is not sufficient.

Global layer

Match the global gauge group, genuine line and surface spectra, charge pairing, discrete theta angles, background counterterms, and partition functions in nontrivial bundles. Two theories with the same local Lagrangian and Lie algebra can differ at this layer: Aharony, Seiberg, and Tachikawa 2013, §§1–2 (PDF) exhibit the distinction through genuine line operators, while Gaiotto, Kapustin, Seiberg, and Willett 2015, §§3–4 (PDF) formulate the associated higher-form symmetries and backgrounds.

Dynamical layer

Match deformations, Higgsing, mass flows, compactification, phases, interfaces, and observables not all fixed by the same protected structure. The evidence page shows how to avoid counting several consequences of one anomaly polynomial or one localization identity as independent confirmation.

A compact worked comparison

Free four-dimensional Maxwell theory illustrates why the layers cannot be collapsed. In the conventions developed on the dualization page, order the source-free field strengths as $(F,G)$, with $F$ tied to the Bianchi identity and $G$ to the equation of motion. The active $S$ generator acts as

$$\begin{pmatrix}F\\ G\end{pmatrix} \longmapsto \begin{pmatrix}0&1\\ -1&0\end{pmatrix} \begin{pmatrix}F\\ G\end{pmatrix} = \begin{pmatrix}G\\ -F\end{pmatrix} = \begin{pmatrix}F_D\\ G_D\end{pmatrix}, \qquad \tau\longmapsto-\frac{1}{\tau}.$$

That calculation alone does not specify a quantum equivalence. One must also state flux quantization, the electric–magnetic lattice, the set of genuine Wilson–’t Hooft lines, boundary conditions, zero modes, and possible spin or nonspin dependence. An integral symplectic transformation preserves the Dirac pairing; a real rotation of classical equations need not preserve the quantum lattice. The dualization page performs this distinction explicitly.

What counts as a failed comparison

A proposed duality needs revision when any required sector gives incompatible, convention-translated data. Typical examples are:

• a genuine line on one side has no image or maps to a non-genuine operator;
• background-field partition functions differ by more than an allowed local counterterm;
• a mass deformation reaches different gapped phases or leaves an unreported topological sector;
• a purported operator violates unitarity and becomes free, but the accidental symmetry is absent from the dictionary;
• two limits used in the comparison do not commute;
• a “new” check is algebraically implied by evidence already used.

A failure can narrow rather than destroy a claim. A full duality may become a protected-subsector equivalence; an exact claim may become infrared-only; a missing topological factor may restore equality after it is included explicitly.

Chapter outputs

After completing this chapter, a reader should be able to produce:

• a precise source and target theory definition;
• a direction-tagged dictionary for parameters, local operators, extended operators, states, and backgrounds, with an inverse only where one is established and with unmapped or emergent entries shown explicitly;
• a named regime and a list of decoupled sectors;
• a transformation rule for gauging, deformations, compactification, and interfaces;
• an anomaly and global-form comparison;
• an evidence-dependency matrix and explicit falsifiers.

These objects are the input to every named duality family later in the volume. They also make clear when two calculations agree because the theories are dual, when they agree because both compute the same protected invariant, and when the available evidence simply does not decide.

Review the chapter

For a duality claim you know, answer all five questions before consulting the literature:

1. Along which axes is it exact, infrared, emergent, operational, or restricted to a subsector?
2. Which global forms and genuine extended operators are included?
3. What is the map of relevant deformations and vacua?
4. Which two checks use genuinely independent inputs?
5. What concrete result would falsify the stated scope?

If any answer is missing, route to the corresponding page above before adding another matching observable.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. DOI. Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open PDF.