Duality Claims, Dictionaries, Regimes, and Evidence

A duality comparison begins with two complete theory definitions and a typed map between them. “The operators match” is not enough: the claim must say which regime is shared, which sectors decouple, how parameters and backgrounds transform, and what evidence is independent. The result is a falsifiable statement rather than a resemblance between formulas.

Required background. Symmetry, redundancy, and duality distinguishes physical equivalence from gauge redundancy, and ’t Hooft anomaly matching supplies a necessary infrared test. Helpful background. Infrared phases and conformal windows illustrates why the regime must be explicit.

Define both theories as physical objects

Write a candidate duality as

$$\mathcal T_A[\mathcal D_A] \quad\longleftrightarrow\quad \mathcal T_B[\mathcal D_B],$$

where $\mathcal D$ includes more than a Lagrangian density. At minimum it contains:

$$\mathcal D=(d,G_{\mathrm{global}},G_{\mathrm{gauge}}, \text{matter},\text{couplings},\text{discrete data}, \text{backgrounds},\text{operator spectrum},\text{boundary data}).$$

The gauge entry is the global group, not only its Lie algebra. The discrete data include theta angles, spin versus nonspin dependence, quotient choices, and local counterterms for background fields. The operator spectrum includes genuine extended operators and their fusion, not merely gauge-invariant polynomials. The way a fixed gauge algebra can support inequivalent global forms and genuine-line spectra is made explicit in Aharony, Seiberg, and Tachikawa 2013, §§1–2.

If either side is defined only as an infrared fixed point, replace a microscopic action by intrinsic CFT data plus a specification of relevant deformations. If one side contains a decoupled free or topological sector, display it:

$$\mathcal T_A\xrightarrow{\mathrm{RG}} \mathcal T_{\mathrm{IR}}\otimes\mathcal T_{\mathrm{dec}}, \qquad \mathcal T_B\xrightarrow{\mathrm{RG}} \mathcal T_{\mathrm{IR}}.$$

Suppressing $\mathcal T_{\mathrm{dec}}$ can preserve selected local correlators while spoiling partition functions, anomalies, or line spectra.

Build a bidirectional dictionary

A dictionary entry is a typed relation, not an unlabeled arrow. For every entry record source, target, quantum numbers, normalization, regime, and possible mixing.

Type	Side A	Side B	Required checks
Parameters	$m_A,g_A,\theta_A,\xi_A$	functions of $m_B,g_B,\theta_B,\xi_B$	dimensions, periodicities, complex conjugations, counterterms
Local operators	$\mathcal O_A$	$c\mathcal O_B+$ mixing	spins, charges, dimensions, OPE and contact terms
Moduli	branch and coordinates	branch and coordinates	singular strata, metrics when claimed, vacuum map
States	charge $\gamma_A$	charge $\gamma_B$	masses, pairings, statistics, chamber
Extended operators	line/surface/defect	line/surface/defect or sum	genuineness, screening, fusion, endpoints
Backgrounds	bundle and connection	transformed background	anomaly inflow, local counterterms, flux sectors

Bidirectionality matters. A proposed map can be injective on a protected ring while missing an entire sector on the target side. A complete equivalence requires an inverse after null operators, gauge identifications, and decoupled factors are treated.

Operator mixing is especially important along RG flow. If several operators share quantum numbers, the infrared primary is generally a linear combination. An operator hitting a unitarity bound can become free and generate an accidental symmetry; the dictionary and anomaly computation must then be enlarged.

Name the regime and precision

Every claim should be expressible as a quantified statement. Examples are:

• equality of complete partition functions on all closed spin four-manifolds with matched background bundles;
• equality of correlators at separated points in the common infrared fixed point;
• equivalence of a $Q$-cohomological operator algebra;
• agreement through order $1/N^2$ in a named large-$N$ limit;
• matching of BPS indices in a specified chamber.

Words such as “exact” and “nonperturbative” are not substitutes for this domain. If equality is only known for protected quantities, say so. If a contact term is scheme-dependent, specify the allowed local counterterm rather than demanding literal equality.

The next page supplies a decision procedure for selecting the strongest justified category.

Separate predictions from correlated checks

Suppose an infrared R-symmetry fixes both operator dimensions and a supersymmetric partition function. Agreement of those two outputs is useful, but they share a decisive input and are not fully independent. Similarly, several anomaly coefficients may all follow from the same fermion charge table.

For each check $C_i$, list its inputs $I(C_i)$. Define an overlap matrix

$$M_{ij}=\frac{|I(C_i)\cap I(C_j)|} {|I(C_i)\cup I(C_j)|}.$$

This numerical measure is only organizational, but it exposes shared assumptions. More important is the logical question: could $C_i$ fail while $C_j$ still passes? If not, they should not be advertised as two independent confirmations.

A strong comparison combines different layers, for example:

1. anomaly matching from ultraviolet charges;
2. a deformation reaching the same gapped phase;
3. an extended-operator and global-form match;
4. an exact observable computed by different weakly coupled descriptions.

Even this collection is evidence for a stated duality, not a universal proof unless a separate theorem establishes equivalence.

Worked example: the structure of Seiberg duality

For $SU(N_c)$ SQCD with $N_f$ flavors in the conformal window, the electric theory is proposed to share its infrared fixed point with a magnetic $SU(N_f-N_c)$ gauge theory containing magnetic quarks $q,\widetilde q$, singlet mesons $M$, and

$$W_{\mathrm{mag}}=M^i{}_j q_i\widetilde q^j$$

up to a matching scale and normalization. A minimal dictionary contains

$$Q^i\widetilde Q_j\longleftrightarrow M^i{}_j,$$

plus baryon maps whose powers of the holomorphic scales fix dimensions and charges. It also contains the parameter map, global symmetry quotient, background contact terms, moduli branches, and the treatment of accidental free fields near the edge of the conformal window.

This dictionary and its holomorphic mass deformations are derived in Seiberg 1995, §§2–4. Anomaly matching and chiral-ring agreement are necessary checks; the general anomaly-matching condition appears in ’t Hooft 1980, §§III.10–III.12, pp. 149–151. Mass-deforming one flavor and recovering the adjacent dual pair is another. Matching a protected index adds information only after conventions, integration contours, and decoupled factors are aligned. None of these checks licenses an exact ultraviolet equivalence: the claim is an infrared duality.

Explicit falsifiers

A comparison is testable when it lists outcomes incompatible with its claimed scope. Examples include:

• an unmatched ’t Hooft anomaly after every allowed local counterterm is included;
• different genuine-line lattices under a claimed exact equivalence;
• a relevant deformation whose two endpoints have inequivalent symmetry-protected topological responses;
• an operator required by invertibility with no target and no null relation;
• different unitary fixed-point central charges after accidental sectors are included.

A failed falsifier test may reveal a missing sector or a claim that was too strong. Revise the definition and repeat all dependent checks; do not retain the original headline while silently weakening its scope.

Common pitfalls

Comparing Lagrangians instead of theories. Field redefinitions, global quotients, counterterms, and line spectra can change the physical interpretation without changing local equations of motion.

Using one protected equality as proof. An index deliberately discards long multiplets. Its equality supports the protected-sector map unless further evidence connects the full theories.

Counting derived consequences separately. Dimensions, R-charges, and parts of a localized observable may share the same extremized R-symmetry. Track their common inputs.

Exercises

A proposed infrared duality matches all continuous anomalies and a supersymmetric index, but side B contains a decoupled $\mathbb Z_2$ topological gauge theory.

1. Which observables can miss the extra sector?
2. Name two comparisons that can detect it.
3. State the strongest safe claim if the sector is omitted from side A.

Solution

Local correlators of operators neutral under the topological sector, and some normalized indices on simple manifolds, can miss it. Ground-state degeneracy on spatial manifolds with nontrivial cycles, genuine-line fusion, partition functions in nontrivial backgrounds, or dependence on topology can detect it. Without matching or explicitly factoring out the $\mathbb Z_2$ theory, one may claim equivalence only of the tested local or protected subsector, not equality of the complete infrared theories.

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. arXiv:1305.0318.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. arXiv:hep-th/9411149.
• ’t Hooft, Gerard. “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking.” In Recent Developments in Gauge Theories, 135–157. Plenum Press, 1980. doi:10.1007/978-1-4684-7571-5_9.