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arXiv:2608.15001

Spinning Conformal Correlators from Neural Networks

Manas Dogra, James Halverson, Joydeep Naskar

Published Aug 15, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.15001

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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.
A bstract We use the embedding formalism to construct conformal fields in D dimensions, by restricting Lorentz-invariant ensembles of homogeneous neural networks in ( D + 2) dimensions to the projective null cone. Conformal correlators may be computed using the parameter space description of the neural network. Exact four-point correlators are computed in a number of examples, and we perform a 4D conformal block decomposition that elucidates the spectrum. In a non-unitary example the decomposition precisely matches OPE coefficients for the self-correlator, but not for the mixed correlator. In others, the analysis is facilitated by recent approaches to Feynman integrals. Generalized free CFTs are constructed using the infinite-width Gaussian process limit of the neural network, enabling a realization of the free boson. The extension to deep networks constructs conformal fields at each subsequent layer, with recursion relations relating their conformal dimensions and four-point functions. Numerical approaches are discussed.
Neural network conformal theories were first introduced in \cite{Halverson:2024axc} and exact two-, three- and four-point correlators were computed from an ensemble of infinite-width neural networks. In particular, the four-point correlator of any scalar operator satisfied crossing-symmetry and conformal block decomposition across different channels were in agreement without any constraint on moments. An infinite tower of degenerate operators with the same scaling dimension were found. In this note, we take a closer look at the structure of correlators and relation between operators. We also layout a recipe to construct the algebra of the theory.
We demonstrate that various aspects of Conformal Field Theory are amenable to machine learning. Relatively modest feed-forward neural networks are able to distinguish between scale and conformal invariance of a three-point function and identify a crossing-symmetric four-point function to nearly 100% accuracy. Furthermore, neural networks are also able to identify conformal blocks appearing in a putative CFT four-point function and predict the values of the corresponding operator product expansions (OPE) coefficients. Neural networks also successfully classify primary operators by their quantum numbers under discrete symmetries in the CFT from examining OPE data. We also demonstrate that neural networks are able to learn the available OPE data for scalar correlation function in the 3D Ising model and predict the twists of higher-spin operators that appear in scalar OPE channels by regression.
We demonstrate that simple feed-forward neural networks (NNs) can accurately compute correlation functions of conformal field theories (CFTs) on a line. Strikingly, by optimising a NN solely on crossing symmetry and providing only the scaling dimension of the leading non-trivial operator and the correlator's value at a single "anchor point", we can reconstruct target physical correlators to within a few percent. We establish the robustness of this minimal-data approach across a broad class of theories and dimensions, including generalised free fields, contact and one-loop Witten diagrams in AdS2_22​, unitary and non-unitary 2d minimal models, the 3d Ising model, and half-BPS correlators in 4d N=4\mathcal{N}=4N=4 super-Yang-Mills theory, together with several thermal two-point functions, notably including those of the 3d Ising model. We argue that this remarkable alignment between NNs and CFTs stems from the spectral bias of gradient-based training, which heavily favours smooth functions. To ground this connection, we analyse the smoothness of conformal correlators using fractional Sobolev semi-norms, Chebyshev spectral decompositions, and a measure based on curvature. Finally, we establish the broader reconstructive power of this technique by extending it beyond the diagonal kinematics of the line.
We consider conformal field theories around points of large twist degeneracy. Examples of this are theories with weakly broken higher spin symmetry and perturbations around generalized free fields. At the degenerate point we introduce twist conformal blocks. These are eigenfunctions of certain quartic operators and encode the contribution, to a given four-point correlator, of the whole tower of intermediate operators with a given twist. As we perturb around the degenerate point, the twist degeneracy is lifted. In many situations this breaking is controlled by inverse powers of the spin. In such cases the twist conformal blocks can be decomposed into a sequence of functions which we systematically construct. Decomposing the four-point correlator in this basis turns crossing symmetry into an algebraic problem. Our method can be applied to a wide spectrum of conformal field theories in any number of dimensions and at any order in the breaking parameter. As an example, we compute the spectrum of various theories around generalized free fields.
We study various questions related to operators with spin in quantum conformal field theory in dimensions higher than two. In particular, we classify conformally-invariant tensor structures which appear in correlation functions of local operators and develop tools for computation of conformal blocks which contribute to these functions. We study the crossing equations for four-point functions using numerical and analytical techniques. Finally, we explore the question of analytic continuation of local operators in spin, which leads us to a simple proof of a generalized Lorentzian inversion formula.