Emergent and Non-Lagrangian N=1 Descriptions

An infrared $\mathcal N=1$ theory need not have a known weakly coupled Lagrangian description. It can still be defined intrinsically by its superconformal symmetry, anomalies, protected operators, deformations, moduli, defects, and correlation data. The evidence must be strong enough to distinguish a genuine QFT from a compatible list of protected numbers, and alternative completions should remain visible whenever unprotected data are unknown.

Required background. Protected SCFT data supplies normalized intrinsic inputs, while exact, infrared, and emergent equivalence fixes the claim category. Helpful background. Duality checks and failure modes calibrates the evidence.

What “non-Lagrangian” means

The term means that no useful weakly coupled Lagrangian frame is known for the theory or regime under discussion. It does not prove that no ultraviolet completion or auxiliary construction exists. Many such theories are defined through compactification, string or brane constructions, strongly coupled limits of gauge theories, or RG endpoints reached from a Lagrangian deformation.

An emergent description emphasizes that the infrared variables or symmetry are absent microscopically. Examples include accidental abelian currents, enhanced supersymmetry, or mutually nonlocal degrees of freedom at a singular point. The emergent theory is not obtained by merely deleting heavy fields; its operator map can be intrinsically nonlocal from the ultraviolet viewpoint.

A minimum intrinsic theory card

Component	Minimum content
Spacetime and algebra	Dimension, superconformal algebra, spin structure, and preserved supercharges
Symmetry	Faithful ordinary and generalized groups, anomalies, higher-group data, and global form
Protected spectrum	Short multiplets with exact quantum numbers, relations, and recombination ambiguities
Central data	$a,c$, flavor levels, normalization, status, and free/topological factors
Moduli and vacua	Branch dimensions, singular strata, coordinate rings, and light sectors
Deformations	Relevant and marginal operators, endpoints, scale maps, and accidental symmetries
Extended objects	Genuine lines, surfaces, boundaries, defects, fusion, and charge lattices where known
Unprotected data	Known correlators, numerical bounds, gaps, or an explicit statement that they are unknown
Construction and evidence	UV flow, compactification, dual frames, exact observables, independent checks, and falsifiers

An anomaly polynomial and an index do not fill the entire card. If extended or unprotected sectors are blank, the claim must be limited accordingly.

Evidence for an RG-defined endpoint

Suppose a known ultraviolet theory is deformed by $\delta W=\lambda\mathcal O$ and is proposed to reach an isolated SCFT $\mathcal T_*$. A controlled identification should establish:

1. the deformation is relevant and preserves the claimed symmetries;
2. no runaway or alternative vacuum captures the flow;
3. all operators hitting unitarity are separated as free fields;
4. a-maximization yields positive, consistent $a,c$ and satisfies $a_{\mathrm{UV}}>a_{\mathrm{IR}}$;
5. protected operator dimensions and anomalies fit the proposed algebra;
6. relevant deformations of $\mathcal T_*$ reach independently understood endpoints;
7. a protected partition function or index agrees after decoupled factors are removed;
8. global forms and defect sectors are compared to the extent claimed.

These conditions can identify a strongly supported endpoint. A theorem of existence would require a nonperturbative construction satisfying QFT axioms or an equivalent mathematical control not usually supplied by anomaly matching alone.

Emergent supersymmetry

An $\mathcal N=1$ flow can exhibit an infrared symmetry consistent with $\mathcal N=2$. The claim is much stronger than finding degenerate operator dimensions. The infrared operators must assemble into $\mathcal N=2$ multiplets, and the $\mathcal N=1$ R-current and an additional flavor current must combine into the $U(1)_r\times SU(2)_R$ symmetry with the appropriate anomaly relations.

For a candidate $\mathcal N=2$ endpoint, one can test the standard anomaly identities

$$\operatorname{Tr}r =\operatorname{Tr}r^3 =48(a-c),$$

and

$$\operatorname{Tr}(rI_3^2)=4a-2c,$$

in conventions where $I_3$ is the Cartan generator of $SU(2)_R$. One must first translate the ultraviolet $\mathcal N=1$ currents into the candidate infrared basis and remove free fields.

Additional evidence includes the required extra supercurrent multiplet, Coulomb-branch operator dimensions compatible with $\mathcal N=2$ special geometry, and an index organizing into $\mathcal N=2$ characters. None of these should be inferred from the central charges alone.

Example: nilpotent N=1 deformations

A useful construction starts from a four-dimensional $\mathcal N=2$ SCFT with flavor moment-map operator $\mu$ and adds an $\mathcal N=1$ chiral multiplet $M$ in the adjoint of the flavor group,

$$W=\operatorname{Tr}M\mu.$$

Giving $M$ a nilpotent expectation value breaks the flavor symmetry and triggers an $\mathcal N=1$ RG flow. The surviving singlets couple to components of $\mu$. One then:

1. determines the unbroken currents and trial R-symmetry;
2. performs a-maximization;
3. removes every operator that violates unitarity and repeats;
4. compares the interacting anomalies and spectrum with a candidate endpoint.

In selected examples studied by Maruyoshi and Song and subsequent work, the interacting endpoint is consistent with an Argyres–Douglas theory—the mutually nonlocal fixed points were introduced in Argyres and Douglas 1995, §§2–4—and enhanced $\mathcal N=2$ supersymmetry. The central-charge, Coulomb-dimension, and index evidence for enhancement is presented in Maruyoshi and Song 2017, pp. 1–5, arXiv PDF, while the broader family of $\mathcal N=1$ deformations is analyzed in Maruyoshi and Song 2017, §§2–5. The conclusion is example-dependent: other nilpotent orbits can produce ordinary $\mathcal N=1$ endpoints, runaways, or additional decoupled sectors.

This construction is pedagogically valuable because it exposes the accidental-symmetry step. Performing a-maximization only once can yield incorrect central charges and obscure the enhancement.

Sparse protected data are not unique

Two distinct SCFTs can share $a$, $c$, global symmetry, and several anomaly coefficients. Even a full superconformal index can fail to distinguish theories differing by sectors invisible to the chosen supercharge. To claim uniqueness, one needs enough data to rule out alternatives, such as:

• different conformal manifolds with the same pointwise anomalies;
• discrete gaugings that preserve local protected operators;
• tensoring with an invertible or topological theory;
• different long-multiplet spectra;
• distinct global forms or line categories.

When uniqueness is unavailable, state a class of candidate completions. Bootstrap bounds on unprotected operator gaps can then discriminate among them without assuming a Lagrangian.

Deformations as fingerprints

Relevant deformations often distinguish theories more sharply than isolated fixed-point numbers. For each protected relevant operator $\mathcal O_i$, record the proposed endpoint of

$$\delta W_i=\lambda_i\mathcal O_i.$$

Compare residual anomalies, vacuum multiplicities, topological phases, and domain walls. If two candidate SCFTs have identical anomalies but one deformation leads to confinement while the other leads to a gapless theory, they are not the same complete QFT.

The deformation graph must include failed and runaway directions. Omitting them biases the identification toward successful matches.

Claim levels

Intrinsic definition: a construction, such as compactification or a mathematically controlled sector, defines the theory independently of a weak Lagrangian.

Strongly supported identification: several independent protected and deformation checks select the same endpoint, but unprotected data remain limited.

Protected-sector match: anomalies, an index, or a chiral algebra agree, with no uniqueness claim for the full QFT.

Candidate endpoint: the data are consistent but alternative completions remain.

These labels can coexist for different portions of one proposal.

Common pitfalls

Defining a theory by anomalies alone. Anomalies are necessary consistency data, not a complete set of observables.

Inferring supersymmetry enhancement from rational dimensions. Extra currents, multiplet organization, and anomaly relations must all be present.

Using “non-Lagrangian” as an evidence claim. It describes the absence of a known useful frame, not a proof that none can exist.

Exercises

A candidate isolated $\mathcal N=1$ SCFT has known $a,c$, flavor anomalies, chiral-ring generators, and a matching index from two UV flows. The global form and line operators have not been studied, and no unprotected correlator is known.

1. What claim is well supported?
2. What full-theory ambiguities remain?
3. Name one useful next test.

Solution

The two flows strongly support the same protected local SCFT sector, provided accidental fields and conventions were aligned. Complete-theory ambiguity remains from discrete gauging, global form, line categories, topological factors, and different long-multiplet spectra. A useful next test is to couple faithful background bundles and compare global responses, classify genuine lines or boundaries, or compute/bootstrap an unprotected four-point observable.

References

• Argyres, Philip C., and Michael R. Douglas. “New Phenomena in $SU(3)$ Supersymmetric Gauge Theory.” Nuclear Physics B 448 (1995): 93–126. arXiv:hep-th/9505062.
• Maruyoshi, Kazunobu, and Jaewon Song. “Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index.” Physical Review Letters 118 (2017): 151602. arXiv:1606.05632.
• Maruyoshi, Kazunobu, and Jaewon Song. “$\mathcal N=1$ Deformations and RG Flows of $\mathcal N=2$ SCFTs.” Journal of High Energy Physics 02 (2017): 075. arXiv:1607.04281.