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Conformal Defects in Neural Network Field Theories

Pietro Capuozzo, Brandon Robinson, Benjamin Suzzoni

TL;DR

The paper develops a formalism to realize conformal defects within Neural Network Field Theories by leveraging embedding-space methods to break ambient conformal symmetry down to defect subgroups. It introduces defect-encoding architectures and defect OPE–like expansions, enabling systematic computation of ambient, defect, and mixed correlators in two toy models. In monomial theories, the defect expansion truncates, yielding exact defect conformal blocks, while in reciprocal theories, correlators require regularization and exhibit richer, meromorphic behavior with positivity constraints. The work extends the NN-FT program to include non-local insertions, offering a pathway to study defect dynamics, defect blocks, and potential extensions to spinning fields, anomalies, and two-dimensional Virasoro structures. These results provide a bridge between conformal defect theory and neural-network realizations, with implications for constructing and analyzing novel conformal field theories via learned architectures.

Abstract

Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in these NN-FTs. We demonstrate this new formalism in two toy models of NN scalar field theories. We develop an NN interpretation of an expansion akin to the defect OPE in two-point correlation functions in these models.

Conformal Defects in Neural Network Field Theories

TL;DR

The paper develops a formalism to realize conformal defects within Neural Network Field Theories by leveraging embedding-space methods to break ambient conformal symmetry down to defect subgroups. It introduces defect-encoding architectures and defect OPE–like expansions, enabling systematic computation of ambient, defect, and mixed correlators in two toy models. In monomial theories, the defect expansion truncates, yielding exact defect conformal blocks, while in reciprocal theories, correlators require regularization and exhibit richer, meromorphic behavior with positivity constraints. The work extends the NN-FT program to include non-local insertions, offering a pathway to study defect dynamics, defect blocks, and potential extensions to spinning fields, anomalies, and two-dimensional Virasoro structures. These results provide a bridge between conformal defect theory and neural-network realizations, with implications for constructing and analyzing novel conformal field theories via learned architectures.

Abstract

Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in these NN-FTs. We demonstrate this new formalism in two toy models of NN scalar field theories. We develop an NN interpretation of an expansion akin to the defect OPE in two-point correlation functions in these models.

Paper Structure

This paper contains 11 sections, 2 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. NN-FTs for conformal fields
  4. Conformal defects in embedding space
  5. Conformal defects in NN-FTs
  6. Examples
  7. Monomial NN-FTs
  8. Reciprocal NN-FTs
  9. Positivity in reciprocal NN-FTs
  10. Conclusion
  11. Useful identities

Key Result

Lemma A.1

Let $\mathcal{N}(0,\hat{\sigma}^2\mathds{1}_{d+2})$ be a multivariate Gaussian distribution centered at the origin with variance matrix $\hat{\Sigma}_{AB}=\hat{\sigma}^2\delta_{AB}$. The expectation value of the defect variables $\hat{\theta}$ then obeys for all $\lambda\in\mathbb{N}^*$. The function $Q(n)$ is defined as counting all possible pairings between $n$ (even) objects

Theorems & Definitions (4)

  • Lemma A.1
  • proof
  • Lemma A.2
  • proof