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arXiv:2512.07946 · Journal of High Energy Physics

Conformal defects in neural network field theories

Pietro Capuozzo, Brandon Robinson, Benjamin Suzzoni

Published May 12, 2026• 1 citation
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Abstract

A bstract Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in NN-FTs by specifying the symmetry-breaking patterns in the network architecture and parameter priors. We demonstrate this formalism through two scalar field toy models. To analyze these theories, we introduce the Neural Product Expansion (NPE); this framework adapts the defect OPE to the probabilistic nature of neural networks, providing a systematic method for decomposing ambient correlation functions into their defect spectra.

DOI
10.1007/jhep05(2026)124

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A bstract Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in NN-FTs by specifying the symmetry-breaking patterns in the network architecture and parameter priors. We demonstrate this formalism through two scalar field toy models. To analyze these theories, we introduce the Neural Product Expansion (NPE); this framework adapts the defect OPE to the probabilistic nature of neural networks, providing a systematic method for decomposing ambient correlation functions into their defect spectra.
Neural Network Field Theories (NN-FTs) typically describe Generalized Free Fields that lack a local stress-energy tensor in two dimensions, obstructing the realization of Virasoro symmetry. We present the ``Log-Kernel'' (LK) architecture, which enforces local conformal symmetry via a specific rotation-invariant spectral prior p(k)∝∣k∣−2p(k) \propto |k|^{-2}p(k)∝∣k∣−2. We analytically derive the emergence of the Virasoro algebra from the statistics of the neural ensemble. We validate this construction through numerical simulation, computing the central charge cexp=0.9958±0.0196c_{exp} = 0.9958 \pm 0.0196cexp​=0.9958±0.0196 (theoretical c=1c=1c=1) and confirming the scaling dimensions of vertex operators. Furthermore, we demonstrate that finite-width corrections generate interactions scaling as 1/N1/N1/N. Finally, we extend the framework to include fermions and boundary conditions, realizing the super-Virasoro algebra. We verify the N=1\mathcal{N}=1N=1 super-Virasoro algebra by measuring the supercurrent correlator to 96%96\%96% accuracy. We further demonstrate conformal boundary conditions on the upper half-plane, achieving 99\% agreement for boundary fermion and boson propagators.
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A bstract Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models M(m,m+1) \mathcal{M}\left(m,m+1\right) M(m,m+1) M m m + 1 . Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.
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 propose a neural-network construction of Euclidean scalar quantum field theories from transformer attention heads, defining nnn-point correlators by averaging over random network parameters in the NN-QFT framework. For a single attention head, shared random softmax weights couple different width coordinates and induce non-Gaussian field statistics that persist in the infinite-width limit dk→∞d_k\to\inftydk​→∞. We compute the two-point function in an attention-weight representation and show how Euclidean-invariant kernels can be engineered via random-feature token embeddings. We then analyze the connected four-point function and identify an "independence-breaking" contribution, expressible as a covariance over query-key weights, which remains finite at infinite width. Finally, we show that summing many independent heads with standard 1/Nh1/N_h1/Nh​ normalization suppresses connected non-Gaussian correlators as 1/Nh1/N_h1/Nh​, yielding a Gaussian NN-QFT in the large-head limit.