Conformal Fields from Neural Networks

TL;DR

This paper introduces a method to construct and analyze conformal fields in various dimensions using neural networks, connecting deep learning models with conformal field theory through the embedding formalism and neural network ensembles.

Contribution

It presents a novel approach to generate conformal fields from neural networks, including exact correlator computations, conformal block decomposition, and the extension to deep networks with recursive relations.

Findings

Exact four-point correlators computed in multiple examples.

Conformal block decomposition reveals the spectrum of the constructed fields.

Neural network ensembles can realize generalized free CFTs and free bosons.

Abstract

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 some examples 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.