Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm

TL;DR

This paper introduces a novel machine learning approach using Grassmannian gradient descent to approximate Ricci-flat Kähler metrics, improving computational efficiency and understanding of Calabi-Yau spaces.

Contribution

It presents a new method combining Grassmannian learning with Donaldson's algorithm for better Ricci-flat metric approximation on Calabi-Yau manifolds.

Findings

Successful implementation on Dwork family threefolds

Observation of nontrivial local minima at different moduli points

Enhanced efficiency in metric computation using Grassmannian learning

Abstract

Motivated by recent progress in the problem of numerical K\"ahler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to K\"ahler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the $h$-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.