Exact, Infrared, and Emergent Equivalence Claims

Different uses of “duality” make different predictions, but their labels are not mutually exclusive boxes. A protected-sector identity can be exact, emergence can qualify an infrared equivalence, and a reformulation can be exact or approximate. Classifying the claim along explicit axes prevents a successful limited check from being promoted into a stronger statement.

Required background. Duality claims and dictionaries supplies the comparison data, and ultraviolet and infrared fixed points supplies the RG language. Helpful background. The 1PI effective action clarifies what an effective description retains.

Classify a claim along five axes

Let $\mathfrak O(\mathcal T)$ denote a specified collection of observables.

Axis	Questions that fix the claim
Scale	All scales in the declared theory, only an infrared endpoint, or a controlled finite window?
Observable scope	Complete QFT, local sector, protected cohomology, selected manifolds, or one named quantity?
Map	Invertible equivalence, direction-tagged operation, quotient, projection, or noninvertible interface?
Error control	Exact, asymptotic, perturbative to a stated order, or bounded by a named expansion?
Emergence	Does the dictionary exist microscopically, only after RG flow, or only in a chosen phase or vacuum?

The familiar labels are useful shorthand only after these axes are filled.

The chapter’s typed dictionary-and-operation map shows where claim scope, one-way operations, evidence, and failed inferences occupy different edges. This page supplies the scale and observable-scope labels for those edges.

Exact equivalence. There is an invertible map $D$ of complete theory data such that

$$\langle\mathcal O_1\cdots\mathcal O_n\rangle_{\mathcal T_A} =\langle D\mathcal O_1\cdots D\mathcal O_n\rangle_{\mathcal T_B}$$

for all allowed manifolds, backgrounds, defects, and scales in the stated domain, up to declared local counterterms. Exact changes of variables and two polarizations of the same quantum system can fall here.

Infrared duality. Two theories have a common long-distance limit,

$$\lim_{E/\Lambda_A\to0}\mathcal T_A =\mathcal T_\ast\otimes\mathcal T_{A,\mathrm{dec}}, \qquad \lim_{E/\Lambda_B\to0}\mathcal T_B =\mathcal T_\ast\otimes\mathcal T_{B,\mathrm{dec}}.$$

The irrelevant directions and ultraviolet spectra can differ. Equality requires the decoupled factors to be matched or explicitly divided out. Seiberg duality supplies a concrete realization with distinct ultraviolet gauge theories and a common infrared description Seiberg 1995, §2, arXiv PDF.

Emergent equivalence. The common infrared description has symmetries, locality, or elementary fields absent microscopically. The equivalence is still an infrared statement, but the word emergent highlights that the dictionary may exist only at the endpoint and need not extend along the flow.

Reformulation. An exact change of variables, Fourier transform, or change of polarization can reformulate the same theory. Separately, two EFT presentations may describe the same light modes only in a controlled kinematic or parameter regime, with errors bounded by $E/M$, $1/N$, or $e^{-S}$. “Reformulation” must therefore carry an exactness and error-control label; it is not automatically a fixed-point claim.

Protected-subsector equivalence. For a differential $Q$, one may have

$$H_Q(\mathcal T_A)\simeq H_Q(\mathcal T_B)$$

as rings, categories, or topological theories. Long multiplets and unprotected correlators lie outside the claim.

An implication hierarchy

With global sectors and allowed backgrounds held fixed, exact equivalence implies agreement of every infrared observable. It also implies agreement after applying a protected or topological functor only when the corresponding supercharge, twist, and functor are identified by the map. Under that hypothesis,

$$\text{exact equivalence} \Longrightarrow \text{same IR theory} \Longrightarrow \text{same selected IR subsector}.$$

Neither arrow can generally be reversed. Many distinct lattice Hamiltonians flow to the same conformal field theory. Different quantum theories can also share an index or chiral ring while differing in long-multiplet spectra or topological sectors.

“Same equations of motion” sits outside this hierarchy. Quantum measures, boundary conditions, operator spectra, and flux sectors can differ even when local classical equations agree.

Tests that discriminate the categories

Question	Exact	Infrared	Protected subsector
Must ultraviolet correlators match?	Yes, after the exact map	No	No
May irrelevant couplings differ?	They must be included in the exact map, possibly through nontrivial reparameterization or redundancy	Yes	Yes
Must genuine extended operators match?	Yes	Yes for the complete IR claim	Only those retained by the subsector
May a decoupled TQFT be omitted?	No	No, unless explicitly factored out	Possibly, if invisible to the stated functor
Does one index establish the claim?	No	No	It can establish only that particular index equality
Are contact terms relevant?	Yes	Universal/fractional parts and allowed schemes matter	Those seen by the subsector matter

To distinguish exact from infrared equivalence, probe finite momentum or deform by an irrelevant operator. To distinguish a complete infrared duality from a local-sector match, probe nontrivial topology, genuine lines, background bundles, and boundary conditions.

RG matching and dangerously irrelevant data

An infrared comparison needs more than a common list of relevant operators. Use the stability convention fixed on the prerequisite page:

$$\mu\frac{d\,\delta\lambda^a}{d\mu} =B^a{}_b\,\delta\lambda^b+\cdots, \qquad B V_I=-\theta_I V_I.$$

Along an eigendirection, $\delta\lambda_I\propto\mu^{-\theta_I}$. An infrared-irrelevant direction has $\theta_I<0$, equivalently a positive eigenvalue of $B$, so it decays as $\mu\to0$. A duality maps the relevant and exactly marginal eigendirections and identifies redundant directions. An irrelevant coupling can nevertheless be dangerously irrelevant when it controls the vacuum structure or becomes necessary after another deformation. It then belongs in the flow dictionary even though it vanishes at the undeformed endpoint.

Accidental symmetries present another obstruction. A UV operator may hit a unitarity bound and decouple as a free field. The correct endpoint is then

$$\mathcal T_{\mathrm{IR}} =\mathcal T_{\mathrm{int}}\otimes\mathcal T_{\mathrm{free}},$$

with a new current multiplet. Comparing only the interacting anomaly coefficients while omitting the free sector gives an incomplete equivalence.

Example: particle–vortex structure versus full equality

In three dimensions, dual descriptions can exchange a particle current with the topological current of a gauge field,

$$j^\mu_{\mathrm{particle}} \longleftrightarrow \frac{1}{2\pi}\varepsilon^{\mu\nu\rho}\partial_\nu a_\rho.$$

This local current map is a central piece of particle–vortex duality. It is not the full statement. One must specify whether gauge fields are compact, which monopole operators exist, spin or spin-$c$ dependence, Chern–Simons contact terms, and the allowed background bundles. The role of these spin and contact-term refinements in three-dimensional dualities is analyzed in Hsin and Seiberg 2016, §§5–6, arXiv PDF. Omitting a level-one invertible spin TQFT can leave separated-point current correlators unchanged while altering partition-function phases and line operators; Kapustin and Seiberg 2014, §§2–3 and 7, arXiv PDF explain how coupling a QFT to a topological sector changes global and extended-operator data.

The safe classification is therefore determined by the most global tested data, not the most striking local formula.

Limit order and apparent equivalence

The following is a schematic warning, not a regulator-independent fermion calculation. Suppose a specified spin QFT depends on energy $E$, circle radius $R$, and a mass $m$. The limits

$$\lim_{mR\to\infty}\lim_{ER\to0} \quad\text{and}\quad \lim_{ER\to0}\lim_{mR\to\infty}$$

need not agree. In a concrete fermionic example one must also fix the starting dimension, spin structure, regulator, and allowed counterterm lattice: only then is an induced lower-dimensional Chern–Simons level meaningful. Compactifying first can retain zero modes or holonomies that the other order removes. A duality obtained in one order is not automatically valid in the other.

Every emergent or effective equivalence should state its hierarchy of scales, for example

$$E\ll R^{-1}\ll m\ll\Lambda,$$

and identify the corrections suppressed at each step.

A classification procedure

1. List the observables and backgrounds claimed to match.
2. State the scale and limit hierarchy.
3. Include free, topological, and boundary sectors.
4. Ask whether the map extends to finite momentum and irrelevant deformations.
5. Ask whether it is invertible on genuine extended operators and superselection sectors.
6. Choose the weakest category that covers every established comparison.

The final step is deliberate. A precise protected-subsector theorem is stronger scholarship than an unsupported claim of full duality.

Common pitfalls

Calling an infrared duality a change of variables. The ultraviolet path integrals can be entirely different and need not be related locally. Their flows, not their microscopic fields, meet.

Dropping an invertible or topological factor. Such a factor may be invisible on flat space but visible on other manifolds, at boundaries, or to extended operators.

Promoting cohomology to the full Hilbert space. $Q$-cohomology deliberately quotients exact states. Equality there leaves long multiplets unconstrained.

Exercises

Two spin theories have identical separated-point correlators at a fixed point and identical local anomalies. With the same framing, spin structure, and background fields, their lens-space partition functions differ by a nontrivial invertible spin-TQFT factor that is not in the predeclared lattice of removable local counterterms.

1. Are the complete fixed-point theories equivalent?
2. Give two valid weaker statements.
3. What extra modification could restore a complete equivalence?

Solution

Not as stated: the global response distinguishes the complete theories. One may claim equality of their local flat-space sector, or equivalence after applying a functor insensitive to the invertible factor. A complete equivalence can be restored by tensoring the appropriate side with the inverse invertible TQFT and then checking that extended operators and boundary data also match.

References

• Hsin, Po-Shen, and Nathan Seiberg. “Level/Rank Duality and Chern–Simons-Matter Theories.” Journal of High Energy Physics 09 (2016): 095. DOI. Open PDF.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 04 (2014): 001. DOI. Open PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open PDF.