Duality Checks, Evidence Independence, Status, and Failure Modes

Duality evidence is heterogeneous. Anomaly matching, chiral rings, deformation flows, partition functions, BPS spectra, extended operators, and numerical observables test different slices of a claim, but many share inputs. A responsible conclusion identifies those dependencies, names the strongest scope actually tested, and lists results that would force revision.

Required background. Duality claims and dictionaries defines the object being tested. Helpful background. Deformation flows and global symmetries and anomalies provide complementary tests.

Status follows scope, not enthusiasm

Use a status only together with a precise claim.

Status	Appropriate use
Exact identity or theorem	A derivation establishes the stated equivalence with explicit hypotheses and global data.
Established conditional result	The conclusion follows if named mathematical or dynamical assumptions hold.
Strongly supported duality	Several materially independent nonperturbative tests cover local and global sectors, with no known counterexample in the stated regime.
Conjectural duality	A coherent dictionary and substantive checks exist, but important sectors or dynamical steps remain unproved.
Protected-sector equivalence	A specified index, cohomology, topological twist, or BPS category agrees; no full-theory conclusion is asserted.
Disfavored or failed proposal	A reproducible mismatch survives convention translation and allowed counterterms.
Open comparison	Required data or calculations are not yet available.

The same pair of theories can carry different statuses for different claims. A protected index identity may be exact while the proposed full infrared duality remains conjectural.

Build an evidence-dependency matrix

Let rows be checks $C_i$ and columns be inputs or assumptions $I_a$. Mark an entry when a check materially uses that input.

Check	Charge table	Operator map	Exact R-symmetry	Localization formula	Global form	RG assumption
Continuous anomalies	✓	✓	sometimes	—	partly	✓
Chiral-ring map	—	✓	partly	—	—	✓
Superconformal index	✓	✓	✓	✓	often	✓
Mass-flow endpoint	✓	✓	partly	—	✓	✓
Genuine-line spectrum	partly	✓	—	—	✓	partly
Unprotected correlator	depends	depends	no	method-specific	depends	✓

This table does not assign mechanical scores. It shows, for example, that an anomaly match and an index match may both rely on the same charge and R-symmetry assignments, while a genuine-line test adds largely different information.

Two checks are logically independent only if one can fail while the other passes within the class of candidate theories. Statistical independence is usually not the right concept because the objects are exact or correlated theoretical predictions.

Necessary conditions and discriminating tests

Anomaly matching is necessary for an infrared duality preserving the symmetry, but many inequivalent theories share the same anomalies. A chiral-ring isomorphism is more detailed, yet it omits long multiplets and global sectors. A supersymmetric partition function may package infinitely many protected operators, but can still be insensitive to a decoupled sector or depend on the same R-charge input as the operator map.

A test is especially discriminating when it is:

• out of sample: not used to construct the dictionary;
• global: sensitive to bundles, topology, or genuine extended operators;
• dynamical: not fixed solely by symmetry or holomorphy already assumed;
• overdetermined: compares a function of several independent parameters, not one number;
• adversarial: chosen where a plausible incomplete dictionary would fail.

Examples include matching a mass flow into a gapped phase with a known topological response, comparing line fusion in different global forms, or reproducing an unprotected observable in overlapping controlled regimes.

A worked dependency analysis

Consider an $\mathcal N=1$ electric–magnetic pair with a proposed operator map and exact infrared R-symmetry.

1. Cubic and mixed anomalies use the fermion charge tables and the proposed symmetry map. They test massless anomaly data but not the full spectrum.
2. Chiral-ring relations use F-terms and the operator map. They test protected multiplication and branches.
3. The superconformal index uses the same R-symmetry and protected charges, plus a localization or trace identity. It greatly enlarges the protected comparison but is correlated with steps 1 and 2; SQCD dual-index identities are developed in Dolan and Osborn 2009, §§3 and 6.
4. A flavor mass flow uses the operator map, then tests a new Higgsing and scale-matching mechanism. It adds dynamical information, as in Seiberg 1995, §4.
5. Global-form and line matching uses charge lattices and screening rather than only the local chiral sector. It tests a substantially different layer; Aharony, Seiberg, and Tachikawa 2013, §§1–2 give explicit four-dimensional examples.

The conclusion should not say “five independent proofs.” It should say that protected local data, a nontrivial RG flow, and global extended data agree, while identifying any untested unprotected sector.

Convention translation before declaring failure

Many apparent mismatches are removable translations. Before treating one as physical, compare:

• generator normalization and charge basis;
• orientation, metric, Hodge star, and Euclidean continuation;
• local background counterterms and contact-term schemes;
• free or topological factors;
• global group and line-operator polarization;
• R-symmetry mixing and accidental currents;
• chamber, contour, and branch choices;
• normalization of composite operators and strong scales.

The allowed translations must be fixed independently. Introducing an arbitrary counterterm or extra sector solely to force agreement makes the claim unfalsifiable.

Failure modes and their meaning

Observed mismatch	First diagnostic	Possible conclusion
Continuous anomaly differs	Translate symmetry basis and free fields	Duality fails if no allowed counterterm or sector repairs it.
Index differs by a simple factor	Check decoupled multiplet or topological factor	Complete claim may need an extra sector.
Local operators match, lines do not	Check global form and polarization	Same local theory but different global theories.
Deformation endpoints differ	Check vacuum, scale matching, and induced terms	Flow square fails or deformation map is incomplete.
Partition functions differ by a phase	Classify quantized local counterterms	Same separated-point theory may use different schemes; a fractional remainder is physical.
Only one chamber matches	Track wall crossing and contour	Claim is chamber-specific, not uniform.
Large-$N$ series matches but finite $N$ fails	Check nonperturbative $e^{-N}$ sectors	Equivalence may be asymptotic only.

A useful negative result states exactly which row failed and which weaker statement survives.

Evidence ceilings

The strongest allowed conclusion is bounded by the most important untested layer.

• Without global form and genuine lines, do not claim equivalence of complete gauge theories.
• Without accidental-symmetry analysis, do not claim exact SCFT operator dimensions near unitarity bounds.
• Without deformation endpoints, do not claim that the dictionary transports relevant operators dynamically.
• With only protected observables, claim a protected-sector match unless independent evidence extends it.
• With a finite parameter or charge window, state that window; do not infer completeness.

These limits are not penalties. They make a claim reusable because later evidence can strengthen a clearly defined statement.

Dated revisions and reproducibility

Research-level comparisons change when new sectors, counterexamples, or exact calculations appear. Record the date, theory definitions, convention choices, sources, and calculation versions behind the current conclusion. A material change—new global form, corrected contact term, altered contour, or added accidental field—requires rechecking every dependent result.

Computational evidence should include frozen inputs, code or algebraic steps, tolerances, and an independent benchmark. Agreement of two programs using the same symbolic identity is one calculation reproduced, not two independent tests.

Common pitfalls

Counting rows instead of dependencies. Ten consequences of one protected identity do not outweigh one global mismatch.

Treating every phase difference as a scheme choice. Only quantized, symmetry-compatible local counterterms are removable.

Preserving a headline after narrowing the result. If a line mismatch leaves only a local-sector equivalence, rename the claim accordingly.

Exercises

A proposed duality passes anomaly matching, chiral-ring matching, and a superconformal-index identity. All three use the same proposed R-symmetry. A mass deformation reaches inequivalent gapped phases whose partition functions differ by a noninvertible TQFT.

1. How many materially distinct layers of positive evidence are present?
2. Can an allowed local counterterm remove the endpoint mismatch?
3. What status is justified for the full infrared theory?

Solution

The positive checks richly test one protected local layer, but they are correlated through the R-symmetry and operator map. The mass flow tests a distinct dynamical/global layer and fails. A noninvertible TQFT is an actual sector, not a local counterterm phase, so it cannot be removed that way. The full infrared duality is disfavored as stated; a protected-sector equivalence may remain valid, or the full claim may be repaired by identifying and matching the missing TQFT before all checks are repeated.

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.
• Dolan, F. A., and Hugh Osborn. “Applications of the Superconformal Index for Protected Operators and $q$-Hypergeometric Identities to $\mathcal N=1$ Dual Theories.” Nuclear Physics B 818 (2009): 137–178. arXiv:0801.4947.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. arXiv:hep-th/9411149.