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Spinning Conformal Correlators from Neural Networks

Manas Dogra, James Halverson, Joydeep Naskar

arXiv:2608.15001v1hep-thcs.LG

TL;DR

Spinning conformal fields require neural-network constructions beyond scalar operators. The paper uses embedding-space architectures to compute spinning correlators and recovers the standard four-dimensional Maxwell field-strength correlator at large N.

  • Problem

    Neural-network constructions of conformal fields had focused on scalar operators, motivating an extension to integer-spin fields and their correlators.

  • Method

    The paper uses reference polarization vectors and homogeneous, transverse embedding-space neural-network architectures to represent spinning primary fields and compute their correlators.

  • Results

    At large N, the ensemble is Gaussian in field strength, has Maxwell-matching two-point functions, and projects to the standard four-dimensional Maxwell correlator.

  • Takeaways & Limitations

    Neural-network architectures and parameter statistics encode conformal tensor structures and associated OPE data in spinning conformal field theories.

  • Takeaways & Limitations

    The Maxwell construction makes the field strength both a descendant and a primary, unlike pure Maxwell theory, where it is not a descendant of a primary.

Abstract

from arXiv · show

We construct spinning conformal fields from neural networks and the embedding formalism, computing their two-, three- and four-point functions in examples, building on scalar conformal field techniques introduced in \cite{Halverson:2024axc}. For a particular ensemble of i.i.d. neurons we recover the 4d Maxwell CFT in the infinite-width limit.

1 Introduction

This work extends neural network constructions of scalar conformal fields to integer-spin fields, using embedding-space methods and reference polarization vectors to compute spinning correlators. The authors construct homogeneous neural network architectures for spinning primaries and evaluate their two-, three-, and four-point functions.

  • Background: Earlier scalar constructions reproduced correct CFT correlators and four-point conformal block decompositions for non-unitary examples with ∆< 0.These results form the immediate foundation for extending the neural network–field theory correspondence to spinning fields.
  • Background: The broader neural network–field theory correspondence encodes field-theory dynamics in neural-network architectures through a statistical partition function.Correlators are obtained by inserting functionals of the neural-network field into the partition function.
  • Contributions: The work extends prior neural-network constructions of scalar conformal fields to integer-spin fields and computes spinning correlators.The construction uses the embedding formalism and reference polarization vectors to represent indices without explicit tensor indices.
  • Contributions: Homogeneous neural network architectures are proposed to represent spinning primary fields.The approach treats neural-network parameters as probabilistic, making the resulting functions probabilistic fields.
  • Contributions: The authors explicitly evaluate two-, three-, and four-point correlation functions for the constructed spinning fields.Different valid CFT tensor structures arise from appropriate permutations of the network parameters.

2 Review of the Embedding Formalism

The embedding formalism linearizes Euclidean conformal transformations as SO(d + 1, 1) actions on an ambient space. Physical fields are recovered on the projective null cone, with spinning operators represented by homogeneous, transverse embedding tensors and index-free polarization polynomials.

  • Embedding space: For Euclidean CFTd with d ≥3, the conformal group SO(d + 1, 1) acts linearly on embedding-space coordinates X = (X+, X−, Xµ).Embedding space is identified with the Lorentz space R^{d+1,1}.
  • Projective null cone: The null-cone constraint X2 = 0 and projective identification X →λX reduce redundant ambient coordinates to light-like rays in the projective null cone.The null-cone restriction reduces the effective dimensionality from d + 2 to d + 1, while points on the same ray are physically equivalent.
  • Spinning fields: A spin-l conformal primary of scaling dimension ∆ lifts to a homogeneous tensor operator ΦA1A2...Al on the null cone with degree −∆.A covariant transversality condition removes redundancies, and physical fields are obtained on a Poincaré section before projection back to Rd.
  • Polarization formalism: Symmetric traceless tensors are encoded index-free as degree-l embedding polynomials by contracting tensor indices with null polarization vectors Z.The shift Z →Z + αX preserves Z2 = 0 and X · Z = 0, leaving the physical field unchanged.
  • Correlator structures: The tensor structures Hij and Vi,jk provide building blocks for spinning correlators, including spin-1 two-point and spin-1–scalar three-point functions.For fixed i, only two of the three Vi,jk structures are independent and may be chosen as a basis.

3 Spinning Conformal Correlators

The section prescribes spinning conformal correlators by enforcing embedding-space homogeneity, Lorentz invariance, finiteness, Z-homogeneity, and transversality. Neural-network constructions reproduce required two- and three-point structures and generate nontrivial four-point tensor dependence, including non-factorizing spin-1 correlators.

  • Construction prescription: The construction requires homogeneity in X, Lorentz invariance, finite correlators, homogeneous degree-l dependence on Z, and transversality X · F = 0.Lorentz invariance is implemented through a SO(d + 2)-symmetric Euclidean theory, Wick rotation, and restriction to the PNC.
  • Neural-network architecture: Shared architecture parameters generate non-trivial correlations between operators with different spins, while the construction reduces to the scalar ansatz for J = 0.The parameter overlaps build on their role in partial OPE matching.
  • Four-point functions: Four-point correlators exhibit the expected H12 and V-type tensor structures with coefficients depending on cross-ratios u and v.The mixed correlator ⟨A(X1)A(X2)Φ(X3)Φ(X4)⟩ factorizes, whereas the four-spin-1 correlator does not, so this architecture is not a GFF sector.
  • Two-point functions: The spin-1 two-point function has the correct H12 tensor structure and scales correctly under Xi → λiXi, while the spin-2 ansatz is the squared vector construction.In d = 2, the spin-2 construction with a dimension-1 Kac-Moody current is reminiscent of a Sugawara stress tensor.
  • Three-point functions: Three-point functions can vanish for specific scalar or mixed-spin choices, but equal-dimension scalar and three-spin-1 correlators yield nonzero CFT-compatible results with pure-number coefficients.The spin-1 three-point coefficients need not vanish, consistent with a generic CFT without obvious symmetries.

4 4d Maxwell CFT as NN-CFT

The construction produces finite-width Maxwell-like NN-CFTs with the correct 4d Maxwell field-strength two-point function, but nonzero connected higher-point correlators. An i.i.d. large-N ensemble suppresses these non-Gaussianities, yielding the local gauge-invariant sector of 4d Maxwell CFT.

  • 4.1 Maxwell-like theories: Finite-width NN-CFTs reproduce the standard d = 4 Maxwell field-strength two-point correlator, but their connected higher-point correlators are generally nonzero.These theories are therefore termed “Maxwell-like.”
  • 4.1 Maxwell-like theories: The finite-width deviations arise because the architecture is nonlinear in Gaussian neural-network parameters, making the resulting theory non-Gaussian.The field strength is simultaneously a primary and a descendant, with the exact redundancy occurring at ∆ = 1.
  • 4.2 Large-N ensemble: N →∞ converts the normalized sum of i.i.d. non-Gaussian channels into a Gaussian field while leaving the two-point function unchanged.This follows from the central-limit theorem in the NNGP limit.
  • 4.2 Large-N ensemble: At large N, field-strength correlators factorize by Wick’s theorem, reproducing the local Maxwell sector; the single-channel coefficient instead differs by a factor of 5/8.The three-point function also vanishes in d = 4 because it contains a factor of (d−4).
  • 4.2 Large-N ensemble: The limits N →∞ and ∆→1 must be taken in that order, because fixed-N connected correlators remain singular as ∆→1 and 1/N suppression otherwise fails.At fixed N, the connected 2n-point function scales as N 1−n(1−∆)−n.

5 Discussions

The work extends neural network constructions of scalar conformal fields to spinning operators, reproducing conformally required tensor structures in two-, three-, and four-point functions. It applies the construction to the local gauge-invariant sector of four-dimensional free Maxwell theory while identifying limitations and directions toward more general CFTs.

  • Contributions: The embedding-space construction imposes the homogeneity and transversality properties required for spinning conformal operators.Its two-, three-, and four-point functions reproduce the tensor structures required by conformal symmetry.
  • Maxwell application: A single network channel reproduces the Maxwell field-strength two-point function but remains non-Gaussian and fails to reproduce the free theory’s higher-point functions.An ensemble of N independent channels suppresses connected non-Gaussian contributions as N →∞ while retaining the field-strength two-point function.
  • Outlook: Parameter sharing and statistics determine which conformally allowed tensor structures are realized in spinning theories.Systematically relating conformal data to network architectures and parameter distributions could support constructing or reverse engineering more general CFTs.
  • Limitations: The analysis is restricted to the local gauge-invariant sector and does not address nonlocal observables such as Wilson lines, ’t Hooft lines, or flux operators.It also leaves unexplored the gauge transformation law of Aµ(x) and extensions to d̸ = 4.

A Calculation Details

The appendix details correlator computations using Schwinger parameterization for the neural-network ansatz at positive scaling dimension. A change of variables isolates a finite integral, while the prescription handles divergences and alternative Wick contractions cover nonconvergent cases.

  • Schwinger parameterization: For Δ > 0, the NN ansatz (3.2) is evaluated using Schwinger parameterization.The parameterization is formal because divergences arise when Θ · X is negative.
  • Schwinger parameterization: Changing variables to u = st and v = s/t decouples the integrals.The divergent v integral represents the Haar measure of the dilatation group and the redundancy of the Schwinger parameterization.
  • Divergences and finite results: The prescription washes out divergences from Θ · X = 0, allowing finite results outside the convergence window in scaling dimensions.After removing the dilatation redundancy, the remaining integral over u is finite.
  • Alternative evaluation: For Δ ≤ 0, Schwinger parameterization does not converge, so the appendix uses the usual Wick-contraction technique instead.For positive Δ, the resulting standard Gamma-function integral includes a normalization factor that can be absorbed by rescaling the scalar field Φ(X).

A.3 Spin-2 2-point correlator

The spin-2 two-point correlator is derived by expanding the squared embedding-space structures and evaluating the resulting Gaussian averages with Wick’s theorem. Null-cone constraints and transversality are then imposed to obtain the simplified contraction structure.

  • Construction: The correlator is built from products of squared antisymmetric embedding-space structures at X1,Z1 and X2,Z2.The structure involves [(Θ·X1)(η·Z1) − (η·X1)(Θ·Z1)]^2 and the analogous factor at point 2.
  • Simplification: The resulting expression combines terms proportional to (X1·X2)^2(Z1·Z2)^2, mixed contractions, and squared cross-contractions.The displayed expansion includes 2(X1·X2)(Z1·Z2)(X1·Z2)(Z1·X2) and (X1·Z2)^2(Z1·X2)^2.
  • Gaussian averaging: Wick’s theorem evaluates the four-Θ and four-η averages as sums of pairwise Kronecker-delta contractions.The derivation uses ⟨ΘAΘBΘCΘD⟩=δABδCD+δACδBD+δADδBC and analogous η contractions.
  • Constraints: Null-cone constraints and transversality, Xi·Zi=0, are crucial for the derivation and its simplification.The text explicitly notes that imposing the null-cone constraints and Xi·Zi=0 is essential.

A.5.1 Expansion in terms of tensor structures

The proposed tensor-structure decomposition is verified by explicitly expanding the H and V structures. Six fractional terms cancel exactly, leaving the stated eight-term polynomial expression for the right-hand side.

  • Expansion procedure: The verification substitutes the explicit definitions of H_ij and V_i,jk into the proposed decomposition.The expansion proceeds term by term through the H V contributions and the product of three V structures.
  • Expansion procedure: The product V1,23V2,13V3,12 expands into eight terms involving contractions among X_i and Z_i.The calculation explicitly enumerates the eight contributions from the three-V product.
  • Cancellation and result: Six fractional terms cancel exactly between the three-V product and the H V expansions, ensuring consistency with the Δ_i = −1 architecture.The architecture contains no fractional term, so this cancellation is required for the decomposition to be consistent.
  • Cancellation and result: The remaining terms combine into the eight-term right-hand side built from pairwise Z_i · Z_j factors and triple Z_i · X_j contractions.The final expression is given across the four displayed result lines.

A.6.1 Factorized correlator

Using equation (3.16), the factorized correlator section computes the four-point correlator.

  • Equation (3.16) is used to compute the 4-point correlator.
  • The passage identifies the calculation as a four-point correlator computation.
  • A.6.1 Factorized correlator: The correlator is computed using the factorized-correlator setup and equation (3.16).

A.6.2 Connected Correlator

Using (3.18), the connected correlator is expanded into four embedding-space contractions and simplified to a sum of four products of inner products involving Z_i and X_i.

  • A.6.2 Connected Correlator: Using (3.18), the correlator expands into four terms involving η1, Θ1, Θ2, Z1, Z2, and X1–X4.The terms differ by the placement of Z1 and Z2 among the contractions with η1, Θ1, and Θ2.
  • A.6.2 Connected Correlator: The first two expanded terms contain (η1 · Z1)(Θ1 · X1)(Θ1 · Z2) and −(η1 · Z1)(Θ1 · X1)(Θ1 · X2)(Θ2 · Z2), respectively.Both retain the common factors (Θ2 · X3)(η1 · X4).
  • A.6.2 Connected Correlator: The simplified result is (Z1 · X4)(X1 · Z2)(X2 · X3) − (Z1 · X4)(X1 · X2)(Z2 · X3) − (X1 · X4)(Z1 · Z2)(X2 · X3) + (X1 · X4)(Z1 · X2)(Z2 · X3).This rewrites the expanded contractions entirely in terms of pairwise inner products among the Z_i and X_i.

A.6.3 Expansion in terms of tensor structures

The section explicitly evaluates the prefactor P for Δ_i = −1 and l = (1, 1, 0, 0), obtaining τ1 = τ2 = 0 and τ3 = τ4 = −1. With term coefficients −1 and 1, the resulting four-point expression simplifies to the form previously obtained in (3.19).

  • Prefactor evaluation: For Δ_i = −1 and l = (1, 1, 0, 0), the parameters are τ1 = τ2 = 0, τ3 = τ4 = −1, and Σ_k τ_k = −2.These values determine the explicit prefactor evaluation.
  • Prefactor evaluation: The prefactor is P = (−2X1 · X2)^−1/3(−2X3 · X4)^2/3(−2X1 · X3)^1/6(−2X1 · X4)^1/6(−2X2 · X3)^1/6(−2X2 · X4)^1/6.It is equivalently written with an overall factor of −2 after extracting powers from the inner products.
  • Tensor-structure expansion: The first and second terms on the right-hand side of (3.20) have coefficients −1 and 1, respectively.Substituting these coefficients and the cross-ratio exponents simplifies the right-hand side and hence G4.

A.6.4 Expansions of other substitutions

The section classifies the nonzero tensor structures generated by parameter relabellings of the neural-network architecture and identifies algebraic dependencies among them. It also finds that the proposed architecture realizes 28 of 43 expected four-point tensor structures, while the resulting correlators are crossing invariant.

  • Parameter relabellings: Only specific parameter relabellings produce nonzero results, yielding G(1), G(2), G(3), G(4), their negatives, and one additional result algebraically dependent on the others.The result identified as G(4) arises from two relabellings, while the final result is a direct algebraic sum of the other three.
  • Parameter relabellings: The architecture never generates the tensor structures V1,23V2,13 and V1,24V2,14, although a different neural-network architecture can obtain them.This limitation is independent of the parameter relabelling performed within the stated architecture.
  • Four-point correlators: 43 expected tensor structures occur in the four-point correlator, of which 28 contribute for the proposed architecture (3.3).The expected structures have schematic forms HH, HV V, and V V V V, and the correlator is expanded using coefficients that depend on cross ratios u and v.
  • Four-point correlators: The four-point functions are crossing invariant by construction and can also be verified explicitly under exchanges such as 1 ↔2.Under 1 ↔2, the first tensor structure maps to the second, and its coefficient transforms consistently from u 4v to u 4.

A.8 Generalized Free Fields

To isolate a true Generalized Free Field sector, the architecture is made linear in parameter space, yielding a Gaussian spin-1 conformal primary to which Wick’s theorem applies directly.

  • Motivation: The bilinear architecture for ∆ = −1 produces non-Gaussian cross-contractions, obstructing strict Generalized Free Field factorization.A true GFF sector therefore requires an architecture linear in parameter space.
  • Construction: A spin-1 conformal primary A(X, Z) of dimension ∆ = −1 is constructed on embedding space R^{d+1,1} from independent Gaussian vector parameters.The construction uses d + 2 independent Gaussian vector parameters and a form factor.
  • Field properties: The resulting field is homogeneous in X with degree −1, linear in Z, and satisfies A(X, X) = 0.These properties encode the required embedding-space homogeneity and transversality.
  • Gaussianity: Because the field is linear in Gaussian parameters, it is itself a Gaussian random field, so Wick’s theorem applies directly to field operators.This Gaussianity enables the desired GFF-sector treatment.

A.8.1 Two-Point Function

The two-point function is expressed as a bilinear combination of embedding-space contractions among X_1, X_2, Z_1, and Z_2.

  • The result is (X1 · X2)(Z1 · Z2) − (X1 · Z2)(Z1 · X2) − (Z1 · X2)(X1 · Z2) + (Z1 · Z2)(X1 · X2).

A.8.2 Three-Point Function … A.9.9 Correlators from “large-N methods”

The sections establish vanishing odd correlators and Wick-factorized even correlators for Gaussian constructions, then develop a large-N neural-network realization whose d = 4 field-strength correlators reproduce Maxwell CFT results. The construction uses a non-primary vector potential, embedding-space antisymmetric structures, and normalization procedures to obtain Maxwell correlators while scalar three-point functions vanish.

  • A.8.2 Three-Point Function: The field’s three-point function vanishes in parity-preserving GFFs because the zero-mean Gaussian distribution has vanishing third moment.The four-point function instead follows Wick’s theorem as H12H34 + H13H24 + H14H23.
  • A.8.3 Four-Point Function: A single Gaussian-parameter architecture produces a GFF at ∆= −1, while generic ∆ motivates the large-N techniques applied to Maxwell theory.The large-N application is developed in Appendix A.9.9.
  • A.9.2 Uplift of field strength correlators to embedding space: The d = 4 Maxwell field-strength two-point function is represented in embedding space using antisymmetric polarization structures, reproducing the physical inversion-tensor contraction.The embedding construction generalizes Hij to two polarizations from different points and projects with an overall factor of −2X1 · X2.
  • A.9.3 Maxwell from non-primary A: The Maxwell construction starts from a deliberately non-primary AM(X), whose anomalous transformation terms vanish in correlators, so F behaves like a primary for correlator calculations.The architecture restores transversality under correlator brackets, recovers the Feynman-gauge propagator, and uses Gaussianity for higher-point functions.
  • A.9.4 ⟨FF⟩from ⟨AA⟩correlator: The two methods for computing ⟨FF⟩ agree: differentiating ⟨AA⟩ produces the expected antisymmetric structure, while the truncated NN calculation yields the same result.Double derivatives eliminate the (X1 · X2)−∆−3 terms through antisymmetry, and the truncated expression is organized as I13(Z2 · Z4) −I14(Z2 · Z3) −I23(Z1 · Z4) + I24(Z1 · Z3).
  • Truncated Ansatz: The truncated and full ansätze use Wick contractions and Schwinger integrals, with the apparent v divergence treated as a redundant Haar-group volume; the truncated calculation reproduces the common ⟨FF⟩ result.The full ansatz sums four contributions and evaluates them through Gaussian source identities, while the truncated result is explicitly identified with the earlier expression.
  • A.9.6 Expressions in terms of H(ZA, ZB; X1, X2) and normalization: The resulting normalization matches Dolan–Osborn and gives the expected Maxwell numerical factors, while large-N factorization reproduces Maxwell’s d = 4 two-point correlator and makes ⟨ΦΦΦ⟩ vanish.The single connected four-channel configuration contains only N terms and is dropped as N →∞; the scalar three-point function vanishes for d = 4.
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