Machine Learning Line Bundle Connections

TL;DR

This paper demonstrates that neural networks can effectively approximate hermitian Yang-Mills connections on line bundles over Calabi-Yau manifolds, using carefully designed loss functions and specific geometric examples.

Contribution

It introduces a machine learning approach to numerically find hermitian Yang-Mills connections on line bundles over complex manifolds, a novel application in geometric analysis.

Findings

Neural networks can approximate hermitian Yang-Mills connections with high accuracy.

The method is demonstrated on elliptic curves, K3 surfaces, and quintic threefolds.

The approach provides a new computational tool for complex differential geometry.

Abstract

We study the use of machine learning for finding numerical hermitian Yang-Mills connections on line bundles over Calabi-Yau manifolds. Defining an appropriate loss function and focusing on the examples of an elliptic curve, a K3 surface and a quintic threefold, we show that neural networks can be trained to give a close approximation to hermitian Yang-Mills connections.