Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm
This work addresses the challenge of computing Calabi-Yau metrics in mathematical physics and string theory, representing an incremental improvement by integrating machine learning into an existing framework.
The paper tackles the problem of numerically approximating Ricci-flat Kähler metrics by introducing a novel approach that combines machine learning, specifically gradient descent on the Grassmannian manifold, with Donaldson's algorithm, applied to the Dwork family of threefolds, resulting in observations of nontrivial local minima as moduli parameters vary.
Motivated by recent progress in the problem of numerical Kähler 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 Kähler 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.