Harmonic $1$-forms on real loci of Calabi-Yau manifolds

TL;DR

This paper uses neural networks to numerically investigate the existence of nowhere vanishing harmonic 1-forms on the real loci of Calabi-Yau manifolds, which could lead to new G2-manifolds and explicit metrics.

Contribution

It introduces a neural network-based numerical approach to approximate Calabi-Yau metrics and harmonic 1-forms on real loci of Calabi-Yau manifolds, exploring their existence.

Findings

Existence of harmonic 1-forms can be ruled out on two manifolds.

Numerical evidence suggests harmonic 1-forms may exist near singular limits on a third manifold.

Potential for a numerically verified proof of harmonic 1-form existence.

Abstract

We numerically study whether there exist nowhere vanishing harmonic $1$-forms on the real locus of some carefully constructed examples of Calabi-Yau manifolds, which would then give rise to potentially new examples of $G_2$-manifolds and an explicit description of their metrics. We do this in two steps: first, we use a neural network to compute an approximate Calabi-Yau metric on each manifold. Second, we use another neural network to compute an approximately harmonic $1$-form with respect to the approximate metric, and then inspect the found solution. On two manifolds existence of a nowhere vanishing harmonic $1$-form can be ruled out using differential geometry. The real locus of a third manifold is diffeomorphic to $S^1 \times S^2$, and our numerics suggest that when the Calabi-Yau metric is close to a singular limit, then it admits a nowhere vanishing harmonic $1$-form. We explain how such an approximate solution could potentially be used in a numerically verified proof for the fact that our example manifold must admit a nowhere vanishing harmonic $1$-form.