From Crossing Solutions to Actual CFTs

Crossing symmetry and positivity are necessary constraints on conformal correlators, but a solution of one truncated four-point equation is not yet a local conformal field theory. The gap is one of quantifiers: an actual CFT must supply a mutually consistent collection of operators and all correlation functions, with locality, positivity, convergence, stress-tensor and global-symmetry data when required, and a reconstruction into a Hilbert space or local net. No general theorem turns a finite numerical crossing solution in dimension greater than two into that complete structure.

Evidence cutoff. 11 August 2026.

Required background. Linear functionals and positivity supplies exclusion logic; existence, construction, and reconstruction claims distinguishes consistent data from a constructed theory. Helpful background. Extremal functionals and spectrum reconstruction supplies candidate-data inference, the Wightman reconstruction theorem states a sufficient full-correlator route, and conformal nets and covariance supplies an operator-algebraic route.

From necessary equations to a conformal theory

Normative question. Which additional consistency, reconstruction, or existence conditions distinguish formal crossing solutions from unitary conformal field theories?

The primary scope is unitary CFT in $d>2$ , where numerical bootstrap usually constrains finitely many four-point functions. Two-dimensional rational, Virasoro, and conformal-net constructions provide important positive examples but have extra algebraic structure. The question does not assume that every generalized conformal theory contains a local stress tensor, nor does it demand a Lagrangian presentation.

A four-point function of identical scalars can be expanded in two channels,

\sum_{\mathcal O}\lambda_{\phi\phi\mathcal O}^{2}F_{\Delta,\ell}(u,v)=0, \qquad \lambda_{\phi\phi\mathcal O}^{2}\ge0,

with the identity and unitarity bounds included. Rattazzi and collaborators showed how positive linear functionals yield rigorous exclusions for the stated equation (Rattazzi et al. 2008). This proves that no CFT with the assumed data can occupy an excluded region, modulo numerical representation errors. It does not prove that every allowed point, kink, or island contains a CFT.

The missing conditions

Layer	Necessary content	Why one crossing solution is insufficient
Operator algebra	One spectrum and OPE coefficients consistent across every operator pair and representation	Independent four-point solutions may not assemble into one associative OPE
Higher-point consistency	Permutation, OPE-channel, and contact-term compatibility for all correlators	Four-point crossing does not supply all higher-point distributions
Positivity and locality	Reflection/Wightman positivity, microcausality, spectral condition, and convergent OPE domains	Positivity for selected squared coefficients is only a projection of full Hilbert-space positivity
Stress tensor and symmetries	A conserved stress tensor with correct Ward identities when “local CFT” requires it; anomaly and global-form data	Generalized free fields can solve crossing without the desired local stress-tensor sector
Reconstruction	Complete Euclidean or Lorentzian data meeting a theorem’s hypotheses, or a constructed conformal net	Finite floating-point data cannot directly satisfy an infinite reconstruction theorem

For an already existing unitary CFT, the operator product expansion converges in a finite domain and its tail is bounded; this justifies systematic bootstrap approximations (Pappadopulo et al. 2012). It is not an existence theorem for arbitrary candidate data. Conversely, a complete family of Schwinger functions obeying Euclidean covariance, symmetry, reflection positivity, regularity, and clustering can be continued and reconstructed (Wightman 1956; Osterwalder and Schrader 1975). The strength of those theorems comes from having all correlators with uniform analytic control.

The strongest “bootstrap completeness” position says that exact crossing, unitarity, and consistency for all correlators should define a CFT even without a Lagrangian. Properly formulated, this is close to a reconstruction program and is compelling. The stronger practical claim—that a finite set of four-point equations and gaps identifies a unique theory—needs additional isolation and existence input. Islands can shrink around data matching a known lattice model or perturbative fixed point, but agreement identifies the candidate only conditionally.

Obstructions and counterexamples to shortcuts

Generalized free fields are consistent crossing solutions and can satisfy Wightman-like axioms, yet often lack a local stress tensor. Large-N crossing admits contact-term ambiguities that change higher-dimension data while preserving low-order constraints. Truncated solutions can contain spurious spectra that drift as derivative order increases. In two dimensions, modular invariance of a torus partition function is necessary but not sufficient to construct all operator products. These are not failures of bootstrap; they show that the target category must be declared.

Assessment. The question is open in general and resolved in selected constructive classes. Full correlator data satisfying a reconstruction theorem are sufficient, and rational or algebraically controlled theories can be built from stronger structures. No accepted finite criterion currently distinguishes every formal $d>2$ crossing solution from an actual local CFT or guarantees that a numerical island contains one.

What would resolve a candidate theory?

For one candidate, resolution could come from a constructive regulator with a proved conformal continuum limit; a conformal-net or vertex-algebra construction with the required locality and positivity; or exact all-correlator data satisfying a reconstruction theorem. For a general criterion, one needs a compactness/existence theorem showing that a convergent hierarchy of certified crossing problems has a limit with OPE convergence, full positivity, locality, and the required stress tensor. A counterexample would be an exact, positive hierarchy that satisfies the proposed finite criterion but provably cannot extend to higher points.

Evidence cutoff and source selection

The finite set joins the founding numerical formulation, OPE-convergence control, reconstruction theorems, and representative constructive CFT frameworks. Targeted arXiv, journal, mathematical-index, and citation-chain searches covered public evidence through 11 August 2026. Numerical islands were not counted as existence proofs unless accompanied by a separate construction.

References

Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. DOI.
Pappadopulo, Duccio, Slava Rychkov, Johnny Espin, and Riccardo Rattazzi. “OPE Convergence in Conformal Field Theory.” Physical Review D 86 (2012): 105043. DOI.
Rattazzi, Riccardo, Vyacheslav S. Rychkov, Erik Tonni, and Alessandro Vichi. “Bounding Scalar Operator Dimensions in 4D CFT.” Journal of High Energy Physics 2008, no. 12 (2008): 031. DOI.
Wightman, Arthur S. “Quantum Field Theory in Terms of Vacuum Expectation Values.” Physical Review 101 (1956): 860–866. DOI.