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Virasoro Symmetry in Neural Network Field Theories

Brandon Robinson

Abstract

Neural Network Field Theories (NN-FTs) can realize global conformal symmetries via embedding space architectures. These models describe Generalized Free Fields (GFFs) in the infinite width limit. However, they typically lack a local stress-energy tensor satisfying conformal Ward identities. This presents an obstruction to realizing infinite-dimensional, local conformal symmetry typifying 2d Conformal Field Theories (CFTs). We present the first construction of an NN-FT that encodes the full Virasoro symmetry of a 2d CFT. We formulate a neural free boson theory with a local stress tensor $T(z)$ by properly choosing the architecture and prior distribution of network parameters. We verify the analytical results through numerical simulation; computing the central charge and the scaling dimensions of vertex operators. We then construct an NN realization of a Majorana Fermion and an $\mathcal{N}=(1,1)$ scalar multiplet, which then enables an extension of the formalism to include super-Virasoro symmetry. Finally, we extend the framework by constructing boundary NN-FTs that preserve (super-)conformal symmetry via the method of images.

Virasoro Symmetry in Neural Network Field Theories

Abstract

Neural Network Field Theories (NN-FTs) can realize global conformal symmetries via embedding space architectures. These models describe Generalized Free Fields (GFFs) in the infinite width limit. However, they typically lack a local stress-energy tensor satisfying conformal Ward identities. This presents an obstruction to realizing infinite-dimensional, local conformal symmetry typifying 2d Conformal Field Theories (CFTs). We present the first construction of an NN-FT that encodes the full Virasoro symmetry of a 2d CFT. We formulate a neural free boson theory with a local stress tensor by properly choosing the architecture and prior distribution of network parameters. We verify the analytical results through numerical simulation; computing the central charge and the scaling dimensions of vertex operators. We then construct an NN realization of a Majorana Fermion and an scalar multiplet, which then enables an extension of the formalism to include super-Virasoro symmetry. Finally, we extend the framework by constructing boundary NN-FTs that preserve (super-)conformal symmetry via the method of images.
Paper Structure (15 sections, 43 equations, 2 figures)

This paper contains 15 sections, 43 equations, 2 figures.

Table of Contents

  1. S1. Uniqueness of the Log-Kernel
  2. S2. Derivation of Neural Modes
  3. Bosonic Modes
  4. Fermionic Modes and Spin Structures
  5. Neveu-Schwarz (NS) Sector
  6. Ramond (R) Sector and the Zero Mode
  7. Ramond (R) Sector and the Zero Mode
  8. S3. Numerical Validations
  9. Central Charge and Normal Ordering
  10. Vertex Operator Spectrum Experiment
  11. Gaussianity and Finite-Width Interactions
  12. S4. Simulation Pseudocode
  13. Algorithm 1: Central Charge (Two-Pass Method)
  14. Algorithm 2: Vertex Operator Scaling
  15. Algorithm 3: Finite-Width Interactions ($G_{4c}$)

Figures (2)

  • Figure S1: Log-log plot of the Neural Vertex Operator two-point function. The slopes correspond to the conformal scaling dimensions $-2\Delta$. The dashed lines indicate the theoretical prediction $\Delta = \alpha^2$ for the free boson, showing excellent agreement with the neural simulation.
  • Figure S2: Scaling of the connected four-point function $G_{4c}$ (interactions) with network width $N$. The blue line represents simulation data ($M=10^8$), which closely tracks the theoretical $1/N$ scaling (red dashed line). The signal is clearly resolvable in the perturbative regime before hitting the statistical noise floor at $N \approx 512$.