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
Equivalence theorem	A construction proves the full stated QFT equivalence, including its declared global data, under explicit hypotheses.
Exact observable identity	A proof establishes equality of one named observable under explicit hypotheses; no wider equivalence follows automatically.
Established conditional result	The conclusion follows if named mathematical or dynamical assumptions hold.
Strongly supported duality	Materially independent tests cover every layer claimed—local, global, and dynamical where included—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 proposal	Evidence weighs against the claim, but no decisive in-scope contradiction has been established.
Failed as stated	A reproducible in-scope mismatch survives convention translation and the predeclared counterterm or factor equivalence.
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.

In the chapter’s typed dictionary-and-operation map, checks use dotted one-way edges and never the double line reserved for a candidate equivalence. The semantic edge list records the same nonimplication without relying on line style.

Build an evidence-dependency matrix

Let rows be checks $C_i$ and columns be inputs, targets, and assumptions $I_a$. A bare check mark hides four different relations, so use:

• C: input used to compute the observable;
• T: dictionary datum being tested;
• H: hypothesis used to interpret agreement as evidence for the claim;
• N: normalization, basis, regulator, or scheme convention.

Check	Charge table	Operator map	Exact R-symmetry	Exact-computation formula	Global form	Common-IR hypothesis
Continuous anomalies	C	T	C/N when an R-current is tested	—	H/T for bundle-sensitive claims	H
Chiral-ring map	N	T	N for graded rings	C: F-terms	—	H
Superconformal index	C	T	C/N	C	H/T	H
Mass-flow composed map	N	T	N	C: threshold relations	T	H
Genuine-line spectrum	C	T	—	C: screening/pairing	C/T	H only for an IR claim
Unprotected correlator	C when charged	T	N if current mixing enters	C: declared method	T when topology enters	H

This table does not assign mechanical scores. It shows, for example, that calculating ultraviolet anomalies uses charge data but no RG assumption; interpreting their match as evidence for a common infrared theory does use one. An anomaly match and an index match can share charge and R-symmetry inputs, while a genuine-line test targets a substantially different layer.

Two checks are logically independent only when neither follows from the other together with the declared shared inputs. A counterexample in which one passes and the other fails proves one nonimplication, not full mutual independence. The dependency record can expose dependence; absence of a listed common input does not prove independence. Statistical independence is usually not the right concept for exact or correlated theoretical predictions.

SQCD example: evidence rows and ceilings

The following compact view applies the four roles to the canonical simply connected $SU(N_c)$ SQCD pair with $N_c\geq3$, where the standard left–right flavor basis applies. The pseudoreal $N_c=2$ theory has enhanced $SU(2N_f)$ flavor symmetry and lies outside this bounded example. Each row states what is tested, what is merely assumed, and how far a positive result can reach. The complete structured matrix supplies the versioned theory cards, dependency graph, source locators, falsifiers, failure propagation, and unresolved work behind every cell.

Check and domain	T: datum tested	C / N: calculation and normalization	H: interpretive hypotheses	Result and confidence ceiling
Continuous anomalies, $N_f\geq N_c+2$	continuous symmetry and charge map	Weyl-fermion tables / $T(\mathrm{fund})=1/2$ and fermion R-charges	mapped symmetry is unbroken; common IR exists	Exact coefficient equality, conditionally established; necessary local protected consistency only.
Displayed chiral data, $N_f\geq N_c+2$	meson, baryon, and F-term map	chiral generators and F-terms / composite and matching-scale conventions	corresponding supersymmetric vacua and strata; common IR exists	Displayed protected generators, charges, maps, and F-term relation agree; not a complete branch-by-branch ring or unprotected match.
Superconformal index	protected charge and fugacity map	electric and magnetic index integrals / measure, contour, and balancing prescription	both integrals compute the same protected trace; common IR exists	Exact equality of the named observable; no full-Hilbert-space or full-QFT conclusion follows.
One-flavor mass flow, $N_f\geq N_c+3$	electric mass to magnetic linear-meson and Higgs map	F-terms, D-flat branch, and threshold exponents / holomorphic-scale convention	all light sectors below both $	m
Endpoint $N_f=N_c+2$	special-color-number repair of the mass-flow map	completely broken magnetic-group instanton / holomorphic-scale convention	the endpoint description is complete	The generic Higgs-only row is excluded; the instanton-corrected endpoint is required, not independent proof of the full duality.
Global backgrounds and extended probes	faithful group, bundles, counterterms, and genuine probes	screening and background analysis / allowed counterterm lattice	the claim includes those global sectors; common IR exists	Not tested in this bounded dossier; this row caps the conclusion at the tested local, protected, and one-flow sectors.

The last row is deliberately not a positive check. Keeping it visible prevents exact local identities from being promoted silently into equivalence of complete gauge theories.

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 normalized protected observable can miss a decoupled sector when its factor is unity in the tested background or was explicitly divided out; a full unnormalized partition function normally multiplies by that sector. Either quantity can also 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 trace and elliptic-hypergeometric identity. Dolan and Osborn prove the identity for the proposed index expressions within stated fugacity, balancing, and contour domains; this is not a theorem that the full Hilbert spaces or QFTs coincide Dolan and Osborn 2009, introduction and §§3, 6, arXiv PDF.
4. A flavor mass flow uses the operator map, then tests a new Higgsing and scale-matching mechanism. It adds dynamical information when it was not used to construct the dictionary Seiberg 1995, §3.2, “Mass terms in the magnetic theory,” arXiv PDF.
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, arXiv v5 PDF.

The conclusion should not say “five independent proofs.” If all five checks have actually been performed and pass, it may say that protected local data, a nontrivial RG flow, and global extended data agree, while identifying any untested unprotected sector. In the bounded SQCD dossier above, however, item 5 is explicitly not tested, so this chapter does not yet establish agreement of the global extended data for that example. The later operator-and-anomaly page and mass-flow page instantiate the protected and dynamical clusters.

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 composed maps differ	Check vacuum, scale matching, induced terms, and operator/background transport	Flow square fails or deformation map is incomplete.
Partition functions differ by a phase	Classify the allowed counterterm lattice for the stated bundles and spin/spin-$c$ structure	Same separated-point theory may use different schemes; a remainder fractional relative to that lattice 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 terms nonperturbative in $1/N$, often $e^{-cN}$ or $e^{-cN^2}$, plus finite-rank and global 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

There is one positive evidentiary layer: the three checks richly test the protected local sector but 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 has failed as stated; a protected-sector equivalence may remain valid. A repaired full claim would have to identify and match the missing TQFT and then repeat every dependent check.

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.
• 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. DOI. Open PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open PDF.