Reconstructing Non-Lagrangian QFTs
Non-Lagrangian theories are not data-free theories. They may be defined by a singular point on a moduli space, a compactification of a higher-dimensional theory, a geometric engineering construction, or an infrared fixed point, and they expose protected spectra, anomalies, defects, and correlation data. These finite packages can identify and sharply constrain a candidate, but no known universal finite package reconstructs an arbitrary interacting QFT uniquely.
Evidence cutoff. 11 August 2026.
Required background. Argyres–Douglas theories, class S, and non-Lagrangian interfaces supplies the main examples; emergent and non-Lagrangian N=1 descriptions supplies RG definitions. Helpful background. Protected N=1 SCFT data and bootstrap export supplies a finite data interface, BPS line and defect observables supplies extended probes, and classification and comparison across QFT frameworks fixes the meaning of equivalence.
How much data specifies a QFT?
Section titled “How much data specifies a QFT?”Normative question. Which operator, defect, anomaly, protected-sector, and consistency data specify a non-Lagrangian QFT, and when is that specification unique?
The scope is principally interacting superconformal theories in four dimensions, where the phrase “non-Lagrangian” means that no useful conventional weakly coupled Lagrangian presentation is known. It does not mean that a Lagrangian is proved impossible, and it does not treat a string or six-dimensional construction as a four-dimensional Lagrangian. Uniqueness always refers to a declared equivalence notion: equality of local QFTs, equality after tensoring with an invertible topological sector, or equality only of a protected subsector are different claims.
Argyres and Douglas identified interacting fixed points with mutually nonlocal massless states in four-dimensional gauge theory (Argyres and Douglas 1995). Gaiotto’s class-S construction organizes broad families by compactifying the six-dimensional theory on punctured surfaces (Gaiotto 2009). These are compelling definitions through an RG or higher-dimensional construction. The associated low-energy Seiberg–Witten geometry, puncture data, and duality frames do not by themselves list every unprotected correlator.
Data packages and their claim ceilings
Section titled “Data packages and their claim ceilings”| Data | What it can establish | Why it need not be unique |
|---|---|---|
| Coulomb-branch dimensions and Seiberg–Witten geometry | Rank, singular loci, low-energy Abelian couplings, BPS central charges | Distinct UV fixed points can share low-energy geometries or differ by decoupled sectors |
| ‘t Hooft anomalies and central charges | Exact RG invariants and strong consistency checks | Coarse invariants are many-to-one |
| Superconformal index and protected spectrum | Short multiplets modulo recombination and chosen fugacities | Long multiplets and many OPE coefficients are invisible; cancellations occur |
| Chiral algebra or topological sector | Infinite protected OPE data and exact correlators in cohomology | The functor discards unprotected operator content and need not be faithful |
| Local and extended defects | Global form, charge lattices, monodromy, boundary categories, and additional OPEs | A sampled defect set may omit sectors; equivalent protected categories need not imply equal bulk dynamics |
| Full correlator family | In principle supports Wightman/Euclidean reconstruction | Infinite analytic and positivity data are required, not a practical finite fingerprint |
Shapere and Tachikawa showed how central charges can be extracted from the topologically twisted low-energy theory under stated assumptions (Shapere and Tachikawa 2008). Beem and collaborators associated a two-dimensional chiral algebra to every four-dimensional SCFT, fixing an infinite protected sector (Beem et al. 2015). Both are powerful reconstruction inputs. Neither paper claims that anomalies plus the chiral algebra uniquely determine the full four-dimensional theory.
Bootstrap constraints can combine protected OPE coefficients with positivity of unprotected channels, as in the chiral-correlator system studied by Lemos and Liendo 2016. This can exclude proposed spectra or isolate islands conditional on gaps and symmetry assumptions. It cannot establish existence merely because a point survives, and it may not distinguish theories that share the bootstrapped correlator. Defect crossing enlarges the tested algebra but inherits the same finite-correlator limitation.
Serious formulations and obstructions
Section titled “Serious formulations and obstructions”There are three strongest notions of specification. A constructive definition gives a UV theory, compactification, or geometric model and proves the desired IR limit. An axiomatic definition gives all correlators or a local net meeting positivity, locality, and covariance. A fingerprint gives finite protected and anomaly data sufficient within a preclassified finite family. Only the third is realistically finite, and its uniqueness is relative to that family.
Known obstructions include decoupled free or topological sectors, discrete gauging, different global forms with identical local correlators, conformal manifolds on which protected data stay fixed, and recombination that erases index information. These supply concrete adversarial tests for any proposed fingerprint.
Assessment. The question is partially resolved for identification within controlled families and open as a universal uniqueness problem. Geometry, anomalies, indices, chiral algebras, defects, and bootstrap data can jointly make an identification compelling and sometimes unique within a stated classification. No theorem shows that a finite protected package specifies an arbitrary non-Lagrangian QFT up to full local equivalence.
What would establish uniqueness?
Section titled “What would establish uniqueness?”A finite-family claim should enumerate the family and prove injectivity of the proposed fingerprint, including discrete quotients and decoupled sectors. A universal reconstruction claim needs either complete correlation distributions satisfying a reconstruction theorem or a local-net construction with all sectors and defects specified. A decisive counterexample is a pair of inequivalent theories sharing the proposed data; a conformal manifold with constant protected data already refutes overly small fingerprints unless the coupling is included.
Related routes are supersymmetry and duality, conformal field theory and bootstrap, analytic and numerical conformal bootstrap, and from crossing solutions to actual CFTs. Boundary and defect bootstrap provides optional depth.
Evidence cutoff and source selection
Section titled “Evidence cutoff and source selection”The finite set covers RG and geometric definitions, anomaly extraction, protected chiral algebras, and bootstrap reconstruction limits. Targeted arXiv, INSPIRE, journal, mathematical-index, and citation-chain searches covered public evidence through 11 August 2026. Matching only a finite index expansion or anomaly vector was never treated as proof of full-theory uniqueness.
References
Section titled “References”- Argyres, Philip C., and Michael R. Douglas. “New Phenomena in SU(3) Supersymmetric Gauge Theory.” Nuclear Physics B 448 (1995): 93–126. DOI.
- Beem, Christopher, et al. “Infinite Chiral Symmetry in Four Dimensions.” Communications in Mathematical Physics 336 (2015): 1359–1433. DOI.
- Gaiotto, Davide. “N=2 Dualities.” Journal of High Energy Physics 2009, no. 8 (2009): 034. DOI.
- Lemos, Madalena, and Pedro Liendo. “Bootstrapping N=2 Chiral Correlators.” Journal of High Energy Physics 2016, no. 1 (2016): 025. DOI.
- Shapere, Alfred D., and Yuji Tachikawa. “Central Charges of N=2 Superconformal Field Theories in Four Dimensions.” Journal of High Energy Physics 2008, no. 9 (2008): 109. DOI.