# Topological Effects in Neural Network Field Theory

arXiv: https://arxiv.org/abs/2604.02313 · HTML: https://mlsanity.com/paper/2604.02313/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 51 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Infinite attention: NNGP and NTK for deep attention networks](https://mlsanity.com/paper/2006.10540/) (2020)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Predicting the outputs of finite deep neural networks trained with noisy gradients](https://mlsanity.com/paper/2004.01190/) (2020)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [A Correspondence Between Random Neural Networks and Statistical Field Theory](https://mlsanity.com/paper/1710.06570/) (2017)
- [Conformal Fields from Neural Networks](https://mlsanity.com/paper/2409.12222/) (2024)
- [Quantum Mechanics and Neural Networks](https://mlsanity.com/paper/2504.05462/) (2025)
- [Fermions and Supersymmetry in Neural Network Field Theories](https://mlsanity.com/paper/2511.16741/) (2025)
- [TASI Lectures on Physics for Machine Learning](https://mlsanity.com/paper/2408.00082/) (2024)
- [Conformal Defects in Neural Network Field Theories](https://mlsanity.com/paper/2512.07946/) (2025)
- [Virasoro Symmetry in Neural Network Field Theories](https://mlsanity.com/paper/2512.24420/) (2025)
- [The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit](https://mlsanity.com/paper/2507.05303/) (2025)
- [Viability of perturbative expansion for quantum field theories on neurons](https://mlsanity.com/paper/2508.03810/) (2025)

## Built on by (most cited first, 2 of 2)

- [Spinning Conformal Correlators from Neural Networks](https://mlsanity.com/paper/2608.15001/) (2026)
- [Neural Network Field Theory at Finite Width](https://mlsanity.com/paper/2608.21588/) (2026)

## Abstract (arXiv)

Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on S1, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.