Spinning Conformal Correlators from Neural Networks

Related work

• The introduction places the construction within a broader NN/FT program spanning Gaussian-process limits, scalar and interacting field theories, fermionic and supersymmetric models, anomalies, and topological theories [2–28].The correspondence originates in the infinite-width Gaussian-process result and has been extended across diverse architectures and field-theory settings.Source: §1 · pp. 2–3

NN/FT correspondence

• The neural-network/field-theory correspondence treats probabilistic network parameters Θ as defining random fields Φ_Θ(X), with architecture encoding dynamics through the parameter distribution and correlator insertions.The mapping reproduces many field-theory symmetries, and under appropriate assumptions every Euclidean QFT admits a neural-network representation with countably infinite parameters [22].Source: §1 · p. 3

Prior scalar construction

• The prior scalar construction [1] used embedding-space homogeneous networks to obtain correct conformal correlators and conformal-block decompositions for non-unitary examples with Δ < 0.Conformal fields are built by restricting rotationally invariant homogeneous ensembles in d + 2 dimensions, then Wick rotating and restricting correlators to the projective null cone.Source: §1 · pp. 2–3

Contribution

• The paper constructs integer-spin conformal fields from homogeneous neural-network architectures using reference polarization vectors Z, enabling index-free spinning primaries and explicit two-, three-, and four-point correlators.Different permutations of network parameters produce distinct valid CFT tensor structures; the construction is applied to the free Maxwell CFT in d = 4.Source: §1 · p. 3

Embedding space

• Conformal transformations act linearly as SO(d + 1, 1) transformations on embedding-space coordinates X = (X+, X−, Xµ).This embedding space is identified with the Lorentz space R^(d+1,1).Source: §2 · p. 4

Projective null cone

• The physical theory is recovered by restricting to the null cone X^2 = 0 and identifying points related by rescaling X → λX, yielding the projective null cone of light-like rays.The null-cone restriction reduces the effective dimension, while projectivization removes the remaining radial redundancy.Source: §2 · p. 4

Spinning fields

• A spin-l conformal primary of scaling dimension ∆ lifts to a homogeneous embedding tensor of degree −∆ subject to a covariant transversality condition.Physical fields are obtained by evaluating embedding fields on a Poincaré-section representative and projecting back to R^d.Source: §2 · p. 4

Polarization formalism

• For symmetric traceless tensors, null polarization vectors produce degree-l embedding polynomials, with transversality realized by the equivalence Z ∼ Z + αX.The architecture of Section 3 realizes this polarization-shift invariance exactly.Source: §2 · p. 5

Correlator structures

• Parity-invariant spinning correlators decompose into theory-specific functions of conformal cross ratios multiplied by tensor structures built from H_ij and V_i,jk.H_ij appears in spin-1 two-point functions, while V_i,jk appears for a spin-1 and two scalar primaries; these structures can be restricted to the Poincaré section.Source: §2 · pp. 5–6

3.1 Principles for Spinning NN-CFT

• The prescription imposes homogeneity in X and Z, Lorentz invariance, finiteness, and transversality, while parameter sharing generates non-trivial correlations between primaries.Lorentzian correlators are obtained by Wick rotating an SO(d + 2)-symmetric Euclidean theory and restricting to the PNC.Source: §3.1 · pp. 6–7

3.2 Concrete Realization

• The ansatz permits scalar correlators to be deduced using the methods of [1], also allows study of J ≠ 0, and defines a section on the projective null cone.The field depends on Z only through the equivalence class Z ∼ Z + αX, rather than through one transverse representative.Source: §3.2 · pp. 7–8

Two-point functions

• The two-point function for spin-1 primaries has the correct H12 tensor structure and scales correctly under Xi → λiXi, with overall scaling λ−∆.Using two fields with independent parameters makes the same-dimension two-point function vanish because both parameter one-point functions vanish.Source: Two-point functions · p. 8

Three-point functions

• Three-point tests reproduce nonzero conformal results for selected scalar and vector correlators, while independent-parameter contractions yield vanishing correlators in the cases required by the construction.The three-vector result expands into four tensor structures with pure-number coefficients, as expected for a conformal three-point function.Source: Three-point functions · pp. 9–10

Four-point functions

• Four-point examples realize conformal tensor structures with cross-ratio-dependent coefficients: mixed spin-1/scalar correlators can factorize, whereas four-vector correlators do not, so the architecture is not GFF.The section does not explicitly compute spin-2 four-point functions, but notes their simplification and exact evaluation in the Maxwell CFT of section 4.Source: Four-point functions · pp. 10–12

4.1 Maxwell-like theories

• Finite-width networks reproduce the Maxwell field-strength two-point function but retain nonzero higher connected correlators, so they are termed “Maxwell-like” theories.Their non-Gaussianity arises because the architecture is nonlinear in Gaussian neural-network parameters.Source: §4.1 · pp. 14, 15, 18
• The vector potential cannot be directly modeled as a Maxwell conformal primary: it transforms anomalously, violates the vector unitarity bound, and has a projector rather than involution tensor structure.The construction instead uses the antisymmetric field strength, which becomes a primary only at Δ=1 after decoupling the singular vector potential.Source: §4.1 · pp. 14, 16

4.2 Maxwell CFT at Large-N

• The normalized sum of N independent channels becomes Gaussian by the central-limit theorem while preserving the two-point function, eliminating single-channel non-Gaussianities.The limits must be taken in order: N→∞ at fixed Δ≠1, followed by Δ→1, because the reverse order is singular.Source: §4.2 · pp. 18, 20
• At N→∞, the ensemble reproduces the local 4d Maxwell sector: field-strength correlators match the Dolan–Osborn normalization, and higher correlators obey Wick factorization.The construction yields vanishing one-point functions and agreement for correlators polynomial in the field strength and its derivatives.Source: §4 · p. 14, §4.2 · pp. 19–20
• The large-N composite scalar Φ := 1/4 F_MN F^MN has the free-Maxwell two-point coefficient and vanishing three-point function in d=4.The single-channel two-point coefficient is instead spoiled by a factor of 5/8, while the three-point function vanishes because it contains a factor of (d−4) [39].Source: §4.2 · p. 20

Contributions

• Neural-network architectures with embedding-space homogeneity and transversality reproduce the conformally required two-, three-, and four-point tensor structures for spinning operators.Correlations between network parameters control which tensor structures and corresponding OPE data appear, so architecture and parameter statistics encode CFT information.Source: §5 · p. 21

Maxwell CFT

• An ensemble of N independent channels approaches four-dimensional Maxwell CFT as N →∞, suppressing connected non-Gaussian contributions while retaining the field-strength two-point function.A single channel reproduces the two-point function but remains non-Gaussian and fails to reproduce the free theory’s higher-point functions.Source: §5 · p. 21

Future directions

• Parameter sharing and statistics determine which allowed conformal tensor structures are realized, motivating systematic maps from network data to CFT OPE data and architecture design for target CFTs.Because higher-point correlators are determined by low-point conformal data, constructing a target theory becomes a constrained factorization problem for lower-point correlators.Source: §5 · p. 21

Limitations

• The construction is limited to the local gauge-invariant sector, omitting nonlocal observables, gauge-potential transformation laws, and extensions to dimensions where free Maxwell theory is not conformal.The paper points to upcoming work as an alternate construction addressing some limitations.Source: §5 · p. 21

A Calculation Details

• Schwinger parameterization computes the NN correlators for ∆ > 0, while Wick contractions handle the nonconvergent ∆ ≤ 0 case.The Schwinger approach yields finite results beyond the scaling-dimension convergence window after regulating divergences; its normalization can be absorbed into a scalar-field redefinition.Source: A Calculation Details · pp. 22–23
• Changing variables to u = st and v = s/t decouples the integrals, with the divergent v integral identified as the dilatation-group Haar measure and modded out.The prescription washes out divergences from Θ·X = 0, leaving a finite u integral.Source: A Calculation Details · p. 22
• For the NN ansatz (3.3), the averaged contraction scales as (X1 · X2)^−∆−1 for generic ∆, with special treatment required for ∆ > 0.This relation is reviewed in section 3 and the ∆ > 0 case is treated separately in Appendix A.1.Source: A Calculation Details · p. 24

Construction

• The correlator is constructed from squared antisymmetric embedding-space structures involving Θ, η, X_i, and Z_i.The basic factors are [(Θ·X_i)(η·Z_i) − (η·X_i)(Θ·Z_i)]^2 for the two insertions.Source: §A.3 · p. 26

Constraints

• Imposing the null-cone conditions and transversality X_i·Z_i = 0 is crucial for the two-point derivation.These constraints are explicitly used alongside the Wick contractions to simplify the correlator.Source: §A.3 · p. 27

Resulting tensor structures

• The resulting expression contains products of embedding-space dot products such as (X_1·X_2)^2(Z_1·Z_2)^2 and mixed terms involving X_1·Z_2 and Z_1·X_2.The expanded terms include the cross structure (X_1·X_2)(Z_1·Z_2)(X_1·Z_2)(Z_1·X_2) and its quadratic counterpart.Source: §A.3 · p. 27

Wick contraction

• Wick’s theorem reduces the Gaussian auxiliary-variable average to pairwise contractions among the Θ and η variables.The derivation uses ⟨Θ_AΘ_BΘ_CΘ_D⟩ = δ_ABδ_CD + δ_ACδ_BD + δ_ADδ_BC and analogous η contractions.Source: §A.3 · p. 27

Expansion in terms of tensor structures

• Substituting the explicit H_ij and V_i,jk definitions and expanding each component verifies the proposed tensor-structure decomposition.The calculation expands the H V terms and the product of three V structures before collecting the surviving terms.Source: §A.5.1 · pp. 30–31
• The six fractional terms from V1,23V2,13V3,12 exactly cancel those from the H V expansions, consistent with the Δ_i = −1 architecture containing no fractional term.The remaining terms reproduce the proposed decomposition’s right-hand side.Source: §A.5.1 · p. 31
• After collection, the right-hand side consists of six Z_i·Z_j terms paired with X contractions and two remaining triple-V contraction terms.The surviving expression is given explicitly across the final collected terms.Source: §A.5.1 · p. 31

A.6.1 Factorized correlator

• Equation (3.16) provides the method for computing the four-point correlator.Source: §A.6.1 · p. 32
• The section applies (3.16) to obtain a four-point correlator.Source: §A.6.1 · p. 32
• The computed object is the factorized correlator discussed in this section.Source: §A.6.1 · p. 32

A.6.2 Connected Correlator

• Applying (3.18) produces four terms involving η1, Θ1, Θ2, Zi, and Xi contractions.The terms alternate signs across the four polarization-contraction structures.Source: §A.6.2 · p. 33
• The expression simplifies to four products of pairwise dot products among Z1, Z2, and X1–X4.The resulting terms have signs +, −, −, +.Source: §A.6.2 · p. 33, §A.6.3 · p. 35

A.6.3 Expansion in terms of tensor structures

• For ∆_i = −1 and l = (1, 1, 0, 0), the exponents are τ_1 = τ_2 = 0 and τ_3 = τ_4 = −1, giving Σ_k τ_k = −2.These values determine the prefactor used in the subsequent expansion.Source: §A.6.3 · p. 33
• The prefactor is evaluated as a product of six embedding-space inner products, with overall coefficient −2 after simplifying the powers.The product contains the factors involving X_1 · X_2, X_3 · X_4, and the four remaining pairings.Source: §A.6.3 · p. 34
• The first and second terms on the right-hand side of (3.20) have coefficients −1 and 1, respectively, enabling the subsequent simplification of G_4.The simplified tensor structures include combinations such as X_12(Z_1 · Z_2) and 2(Z_1 · X_2)(Z_2 · X_1).Source: §A.6.3 · pp. 34–35
• The tensor-structure expansion simplifies to the expression obtained in (3.19), using X_ij = −2X_i · X_j.The final algebraic form is explicitly identified as the result of (3.19).Source: §A.6.3 · p. 35

• Parameter relabellings produce G^(1), G^(2), G^(3), and G^(4), together with their negatives, and these are the only relabellings yielding nonzero results.The relabellings yielding G^(1) and G^(2) also yield their exact negatives, while additional substitutions generate G^(3) and G^(4).Source: §A.6.4 · pp. 35–36
• G^(4) is a direct algebraic sum of the other three structures.The corresponding relabelling produces the negative of (A.32), which is algebraically dependent on the preceding results.Source: §A.6.4 · p. 36
• The architecture never produces the tensor structures V1,23V2,13 and V1,24V2,14, although a different neural-network architecture can obtain them.This limitation holds under all parameter relabellings of the stated architecture.Source: §A.6.4 · p. 36
• 28 of 43 expected four-point tensor structures contribute to the correlator computed from the proposed architecture.The structures have coefficients that are functions of cross ratios u and v.Source: §A.6.4 · pp. 36, 38
• The computed four-point functions are crossing invariant by construction, with the 1 ↔ 2 exchange explicitly mapping the first tensor structure and coefficient to the second.Under the exchange, the coefficient transforms from u^4v to (u/v)^4v = u^4, matching the second term.Source: §A.6.4 · pp. 39–40

A.8 Generalized Free Fields

• The main bilinear architecture can produce non-Gaussian cross-contractions, obstructing strict Generalized Free Field factorization in general.A linear parameter-space architecture is introduced to isolate a true GFF sector.Source: §A.8 · p. 40
• The construction uses d + 2 independent zero-mean, unit-covariance Gaussian vectors to define a spin-1 conformal primary A(X, Z) of dimension ∆ = −1.The Gaussian parameters are contracted with an embedding-space form factor through a linear combination.Source: §A.8 · p. 40
• The resulting field is homogeneous in X with degree −1, linear in Z, and satisfies transversality A(X, X) = 0.Its linear dependence on Gaussian parameters makes it a Gaussian random field, allowing Wick’s theorem to apply directly to field operators.Source: §A.8 · p. 40

A.8.1 Two-Point Function

• The result is (X1 · X2)(Z1 · Z2) − (X1 · Z2)(Z1 · X2) − (Z1 · X2)(X1 · Z2) + (Z1 · Z2)(X1 · X2).Source: §A.8.1 · p. 41

A.8.2 Three-Point Function

• For a parity-preserving GFF, the field’s self three-point function vanishes because the underlying zero-mean Gaussian distribution has vanishing third moment.Source: §A.8.2 · p. 41

A.8.3 Four-Point Function

• Wick’s theorem gives the four-point function as H12H34 + H13H24 + H14H23, and the linear single-Gaussian-parameter architecture realizes a GFF at ∆= −1.For generic ∆, the appendix points to large-N techniques discussed in [1] as the straightforward GFF construction.Source: §A.8.3 · p. 42

A.9.1 Review of some results from CFT

• CFT representation theory shows that a level-1 two-form descendant of a vector has vanishing two-point correlators at ∆=1, enabling Maxwell correlators after rescaling by 1 −∆.The descendant being primary also forces ∆=1; taking ∆→1 then yields Maxwell up to a numerical factor.Source: §A.9.1 · p. 44

A.9.2 Uplift of field strength correlators to embedding space

• The Maxwell field-strength two-point function is lifted to embedding space as an antisymmetric combination of H polynomials, reproducing the d = 4 pure Maxwell correlator.Two independent polarization vectors per point generalize the standard embedding tensor structure, with projection using an overall factor of −2X1 · X2.Source: §A.9.2 · pp. 45–46

A.9.3 Maxwell from non-primary A

• The non-primary architecture breaks transversality to reproduce the Feynman-gauge Maxwell propagator, while correlator transversality is restored through Gaussian averaging.Its anomalous conformal-transformation term vanishes for the chosen architecture and neural-network parameter distribution, so F behaves as a primary inside correlators.Source: §A.9.3 · pp. 47–49

A.9.4 ⟨FF⟩from ⟨AA⟩ correlator

• The ⟨FF⟩ calculation from ⟨AA⟩ uses antisymmetric derivatives and cancellations among permutation terms to obtain the field-strength two-point correlator.The potentially leading (X1 · X2)^−∆−3 contribution cancels, while the remaining tensor structures carry factors proportional to ∆−1.Source: §A.9.4 · pp. 51–52

A.9.4 ⟨FF⟩from ⟨AA⟩correlator

• The ⟨FF⟩ correlator is obtained from the ⟨AA⟩ correlator through a linear combination of differential operators D13, D14, D23, and D24.The displayed expression combines the four differentiated ⟨AA⟩ correlators with alternating signs.Source: §A.9.4 · p. 52

Truncated Ansatz

• The truncated ansatz gives ⟨F1F2⟩=I13(Z2 · Z4)−I14(Z2 · Z3)−I23(Z1 · Z4)+I24(Z1 · Z3), reproducing equation (A.78).The two methods therefore yield the same ⟨F(X1, Z1, Z2)F(X2, Z3, Z4)⟩ correlator.Source: Truncated Ansatz · pp. 53, 56
• The truncated calculation encounters a divergent relative-scaling integral dv/v after imposing the PNC constraint, interpreted as a Schwinger-parameter gauge volume and removed when extracting the finite piece.The integrand depends only on the product of Schwinger parameters, while v is their relative scaling; the ambiguity can be fixed through field normalization.Source: Truncated Ansatz · p. 55

Full ansatz calculation

• The full ansatz expands the field-strength product into four terms T1, T2, T3, and T4, averages Θ and η by Wick contractions, and evaluates the remaining ˜Θ averages with Schwinger parameterization.The calculation uses the untruncated network field and proceeds term by term, beginning with T1.Source: Full ansatz calculation · pp. 56, 58, 60
• The full ansatz yields the complete field-strength two-point correlator in terms of S and Kdiff, matching Eq. (4.14) up to an overall normalization.The calculation uses Gaussian averaging, Schwinger parameters, PNC constraints, and formally removes redundant Schwinger-parameter divergences.Source: Full ansatz calculation · p. 66

A.9.6 Expressions in terms of H(ZA, ZB; X1, X2) and normalization

• The correlator’s tensor structure is rewritten as the antisymmetrized product H(Z1, Z3; X1, X2)H(Z2, Z4; X1, X2) − H(Z1, Z4; X1, X2)H(Z2, Z3; X1, X2), reproducing (A.52) up to numerical factors.Quartic (X · Z) terms cancel when the swapped H products are subtracted, leaving Kdiff.Source: §A.9.6 · pp. 66–67

Normalization match

• The normalization choice produces the Maxwell field-strength correlator and matches the Dolan–Osborn normalized result.The resulting expression has the H-product numerator divided by 16π2 (X1 · X2)4 before the stated cancellations.Source: Normalization match · p. 68

A.9.7 ⟨ΦΦ⟩from truncated ansatz

• The truncated ansatz gives the expected Maxwell field-strength two-point correlator with correct numerical factors, while its one-point function formally diverges and is handled by normal ordering.The scalar architecture evaluates independent Gaussian moments using Wick’s theorem and Schwinger parameterization; divergent pieces are ignored in subsequent computations.Source: §A.9.7 · pp. 68–70
• In d = 4, the truncated ansatz gives a vanishing ⟨ΦΦ⟩, unlike pure Maxwell theory.If higher-point correlators also vanished, the theory would describe a TQFT, although the text states this is unlikely.Source: §A.9.7 · p. 71

A.9.8 ⟨ΦΦ⟩from primary F

• For a primary F, introducing a preferred null vector ¯X and the effective metric K restores strict transversality without changing F correlators.The construction is equivalent to raising with K_MN of Costa et al. [29], and tracing with K is equivalent to tracing the explicitly projected tensor with η_MN.Source: §A.9.8 · pp. 72–73

2-point function

• Even after making F primary, the NN ⟨ΦΦ⟩ remains off the Maxwell result by a factor of 5 because of missing (d + 1)(d + 2) terms.The missing terms are attributed to non-Gaussianities that the subsequent large-N technique suppresses.Source: 2-point function · pp. 77–78

A.9.9 Correlators from “large-N methods”

• The large-N construction uses independent, zero-mean Gaussian channels whose summed field approaches a Gaussian random field as N →∞.Because F is linear in A, F inherits the same large-N statistics and its correlators factorize; normal ordering removes coincident self-contractions for quadratic operators.Source: §A.9.9 · pp. 78–79
• The large-N ensemble enforces Wick factorization and makes the ⟨ΦΦ⟩ correlator match Maxwell by suppressing single-channel non-Gaussianities.Taking infinitely many i.i.d. channels yields Gaussian statistics by the central limit theorem; connected all-channel contributions scale as N rather than N^2 and are dropped.Source: §A.9.9 · pp. 78–80
• The two-point correlator exactly reproduces the Maxwell CFT result for d = 4.Source: §A.9.9 · p. 82
• The three-point correlator is evaluated by applying Wick’s theorem to Gaussian field strengths, yielding eight equal terms that are simplified using the Maxwell two-point correlator.Source: §A.9.9 · p. 82
• The three-point function vanishes for d = 4, because the final expression contains a factor that vanishes in four dimensions.The proportionality constant is dimension independent, and the denominator is non-zero.Source: §A.9.9 · p. 84