Realizing other primary fields of the 2d critical Ising CFT in NN-FTs

Construct neural network field theory architectures within the NN-FT framework that realize the remaining local primary operators of the two-dimensional critical Ising conformal field theory—such as the spin field σ and the energy density ε—beyond the Neural Majorana Fermion, and determine whether these constructions demonstrate that neural networks can naturally represent minimal models with non-trivial operator statistics.

Background

The paper introduces Neural Network Field Theories (NN-FTs) that realize Virasoro and super-Virasoro symmetry in two dimensions, constructing explicit architectures for a free boson (via a Log-Kernel network) and a free Majorana fermion (via Grassmann-weighted features forming the Cauchy kernel). The Neural Majorana Fermion is shown to reproduce the c=1/2 free fermion, identifying it with the fermionic sector of the 2d critical Ising model.

However, the full Ising minimal model includes additional primary fields beyond the Majorana fermion, notably the spin σ and energy ε operators with non-trivial operator statistics. Demonstrating NN-FT realizations of these primaries would provide evidence that neural architectures can encode rational minimal models, extending the current results beyond free fermionic sectors to the full operator content of an interacting rational CFT.