Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918380274974720 |
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| author | Ek, Carl Henrik Kim, Oisin Mishra, Challenger |
| author_facet | Ek, Carl Henrik Kim, Oisin Mishra, Challenger |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11284 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm Ek, Carl Henrik Kim, Oisin Mishra, Challenger High Energy Physics - Theory Machine Learning Algebraic Geometry Differential Geometry 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. |
| title | Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm |
| topic | High Energy Physics - Theory Machine Learning Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2410.11284 |