Virasoro Symmetry in Neural Network Field Theories

TL;DR

This paper introduces the Log-Kernel architecture for neural network field theories that enforces local conformal symmetry, enabling the emergence of Virasoro and super-Virasoro algebras with high accuracy.

Contribution

The authors develop a novel spectral prior and architecture that realize Virasoro symmetry in neural networks, extending to superconformal symmetry with boundary conditions.

Findings

Central charge measured at approximately 1, matching theoretical predictions.

Super-Virasoro algebra verified with 96% accuracy in supercurrent correlator.

Boundary conditions on the upper half-plane achieved 99% agreement in propagators.

Abstract

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 $p(k) \propto |k|^{-2}$. 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 $c_{exp} = 0.9958 \pm 0.0196$ (theoretical $c=1$) and confirming the scaling dimensions of vertex operators. Furthermore, we demonstrate that finite-width corrections generate interactions scaling as $1/N$. Finally, we extend the framework to include fermions and boundary conditions, realizing the super-Virasoro algebra. We verify the $\mathcal{N}=1$ super-Virasoro algebra by measuring the supercurrent correlator to $96\%$ accuracy. We further demonstrate conformal boundary conditions on the upper half-plane, achieving 99\% agreement for boundary fermion and boson propagators.