Conformal Fields from Neural Networks

TL;DR

This work introduces a novel route to conformal field theories by leveraging the embedding formalism and neural-network ensembles. Conformal fields in $D$ dimensions arise from restricting homogeneous Lorentz-invariant networks in $\mathbb{R}^{D+1,1}$ to the projective null cone, allowing conformal correlators to be computed from parameter-space averages. The authors obtain exact 4-point functions in several examples, perform a 4D conformal block decomposition to elucidate the spectrum, realize generalized free CFTs via the infinite-width Gaussian-process limit, and show how deep networks yield recursive conformal fields across layers. They also explore non-unitary solvable models, unitary-inspired amplitude techniques, non-local quarticities, and numerical prospects, outlining future directions for incorporating unitarity, spinning fields, and numerical CFT data extraction.

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.

Figures (1)

• Figure 1: A construction technique pursued in Section \ref{['sec:NNCFT']}, where Lorentzian correlators are obtained via Wick rotation of analytically computed correlators of a rotationally invariant theory on $\mathbb{R}^{D+2}$. See Section \ref{['sec:other_treatments']} for other treatments of the Lorentzian theory.