Emergent and Non-Lagrangian N=1 Descriptions

An infrared $\mathcal N=1$ theory need not have a known weakly coupled Lagrangian description. A compactification, controlled limit, or RG construction may define a candidate endpoint, while anomalies, protected operators, deformations, defects, and correlation data characterize and test it. A finite protected fingerprint is not by itself a complete definition or a uniqueness theorem. The evidence claim must remain narrow enough that alternative completions are visible whenever global or 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, auxiliary description, or mathematically controlled construction exists. Strongly coupled limits of gauge theories, compactifications, and RG endpoints reached from Lagrangian deformations can all supply nonperturbative definitions or evidence of different strength.

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

Three logically different objects should not be conflated:

1. a construction, which specifies how the candidate QFT is obtained;
2. a theory card, which records the observables and structures presently known; and
3. an identification claim, which states how strongly those data select a named endpoint.

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 or 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 or compactification, exact observables, independent checks, evidence cutoff, and falsifiers

An anomaly polynomial and an index do not fill the entire card. A blank field should be marked unknown or not tested, not silently omitted. If the extended or unprotected sectors are unknown, the claim is about a local protected sector rather than a unique complete QFT.

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 asks whether:

1. the deformation is relevant and preserves the claimed symmetries;
2. the chosen vacuum or basin is specified and competing runaway or gapped branches are excluded to the stated extent;
3. every operator hitting unitarity is separated as a free field and a-maximization is repeated;
4. the total ultraviolet and infrared sectors obey $a_{\mathrm{UV}}>a_{\mathrm{IR}}$ for a nontrivial unitary flow;
5. the interacting $a,c$, protected 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 backgrounds, contours, contact terms, global sectors, and decoupled factors are aligned; and
8. global forms and defect sectors agree to the extent included in the claim.

The four-dimensional a-theorem supplies the monotonicity test in item 4 Komargodski and Schwimmer 2011, §§2–3. It must be applied to the complete endpoint, not to the interacting factor on one side and the total theory on the other.

Passing these tests can give a strongly supported identification. A theorem of existence or uniqueness requires a nonperturbative construction with correspondingly stronger control; anomaly matching alone does not supply it.

Testing emergent N=2 supersymmetry

An $\mathcal N=1$ flow can have an endpoint consistent with $\mathcal N=2$. Degenerate dimensions or rational central charges are not sufficient. The operators must organize into $\mathcal N=2$ multiplets, and the $\mathcal N=1$ R-current and an additional flavor current must combine into $U(1)_r\times SU(2)_R$.

With $I_3$ normalized to have eigenvalues $\pm1/2$ in an $SU(2)_R$ doublet, the standard $\mathcal N=1$ subalgebra uses

$$R_{\mathcal N=1}=\frac13r+\frac43I_3.$$

The $\mathcal N=2$ anomaly identities are

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

and

$$\operatorname{Tr}(rI_aI_b) =\delta_{ab}(4a-2c).$$

These conventions and identities are fixed in Shapere and Tachikawa 2008, §2.1, eqs. (9) and (12). Before applying them, translate the ultraviolet $\mathcal N=1$ currents into the proposed infrared basis and remove every free multiplet.

Independent evidence includes an extra supercurrent multiplet, Coulomb-branch dimensions compatible with $\mathcal N=2$ special geometry, deformation endpoints, and an index organizing into $\mathcal N=2$ characters. None follows from $a$ and $c$ alone.

Worked evidence card: the D4-to-H0 flow

Maruyoshi and Song provide a useful bounded example. The starting object is the rank-one $D_4$ SCFT, realized by four-dimensional $\mathcal N=2$ $SU(2)$ gauge theory with four fundamental hypermultiplets and $SO(8)$ flavor symmetry. Add an $\mathcal N=1$ chiral multiplet $M$ in the 28-dimensional adjoint representation of $SO(8)$ and couple it to the moment map $\mu$:

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

Then give $M$ the nilpotent expectation value associated with the principal embedding $SU(2)\hookrightarrow SO(8)$. The general construction is stated in Maruyoshi and Song 2017, §1, eqs. (1)–(2); the $D_4$ example and its post-Higgsing Lagrangian are given in Maruyoshi and Song 2017, §4.1, eqs. (62)–(66).

Card field	Bounded result
Construction	Principal nilpotent $\mathcal N=1$ deformation of the $D_4$ theory
Surviving singlets	Components $M_{j,-j}$ with $j=1,3,5,3'$ after $\mathbf{28}\to V_1\oplus V_3\oplus V_5\oplus V_3$
Accidental-field analysis	The first extremum sends the ultraviolet Coulomb operator and $M_{1,-1}$ below the bound; the source repeats a-maximization twice
Interacting endpoint	Strongly supported identification with $H_0=(A_1,A_2)$
Central data	Interacting-sector $a=43/120$ and $c=11/30$
Coulomb generator	$M_{5,-5}$ has $\Delta=6/5$
Index evidence	After the decoupled chiral factors are removed, the interacting-factor index computed from the resulting $\mathcal N=1$ description agrees with the $H_0$ index
Free factor	Present and essential; this bounded card does not claim a complete enumeration of the ultraviolet fields that decouple along the flow
Unresolved full-theory data	Complete line and defect sectors, global-form comparison, generic unprotected correlators, and a uniqueness theorem
Claim ceiling	Strongly supported identification of the interacting local SCFT factor, not an axiomatic existence or uniqueness proof

The central charges and dimension are the $H_0$ values quoted in Maruyoshi and Song 2017, §4.1, eq. (61), together with the $D_4$ conclusion following eq. (63). The same discussion records repeated decoupling and the index calculation. Because the checked source excerpt does not enumerate the complete free factor in one frozen list, this page does not manufacture one. A complete total-theory anomaly comparison must return to the source calculation and record every decoupled chiral explicitly.

The card has several falsifiers: failure of the repeated anomaly calculation, a residual operator below unitarity, absence of the extra $\mathcal N=2$ current multiplet, an index mismatch after all free factors are removed, or a deformation endpoint incompatible with $H_0$. Agreement of $a,c$, and $\Delta=6/5$ is important but does not make those tests redundant.

Sparse protected data are not unique

A finite list of $a,c$, global symmetry, and anomaly coefficients need not determine a unique SCFT. Even a full superconformal index can miss sectors invisible to the chosen supercharge. Candidate completions may differ by:

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

When uniqueness is unavailable, state the remaining class of completions. Bootstrap bounds may exclude or distinguish some alternatives only after the external correlator, symmetry representation, positivity assumptions, and any gap hypotheses have been specified.

Deformations as fingerprints

Relevant deformations can 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 candidates have identical local anomalies but one deformation confines while the other remains gapless, they are not the same complete QFT. Failed directions and runaways belong in the deformation graph whenever they have actually been established; they should not be inferred from a successful nilpotent-flow example.

Claim levels

Intrinsic construction: a compactification, controlled limit, or mathematical construction specifies the theory independently of a weak Lagrangian, with its own hypotheses stated.

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

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

Candidate endpoint: the available data are consistent, but substantial alternatives remain.

Different portions of one proposal may warrant different labels. Current evidence and unresolved reconstruction questions are maintained in the non-Lagrangian QFT reconstruction dossier and the Supersymmetry and Duality Research map.

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, anomaly relations, and deformation or index evidence must agree.

Using “non-Lagrangian” as an evidence claim. It describes the absence of a known useful frame, not proof that no frame 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, backgrounds, 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 compare faithful background bundles and global responses, classify genuine lines or boundaries, or formulate a specified unprotected four-point bootstrap problem.

References

• Komargodski, Zohar, and Adam Schwimmer. “On Renormalization Group Flows in Four Dimensions.” Journal of High Energy Physics 12 (2011): 099. arXiv:1107.3987.
• 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.
• Shapere, Alfred D., and Yuji Tachikawa. “Central Charges of $\mathcal N=2$ Superconformal Field Theories in Four Dimensions.” Journal of High Energy Physics 09 (2008): 109. arXiv:0804.1957.