Harmonic $1$-forms on real loci of Calabi-Yau manifolds
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916934982828032 |
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| author | Douglas, Michael R. Platt, Daniel Qi, Yidi Barbosa, Rodrigo |
| author_facet | Douglas, Michael R. Platt, Daniel Qi, Yidi Barbosa, Rodrigo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19402 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic $1$-forms on real loci of Calabi-Yau manifolds Douglas, Michael R. Platt, Daniel Qi, Yidi Barbosa, Rodrigo Differential Geometry High Energy Physics - Theory 32Q25 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. |
| title | Harmonic $1$-forms on real loci of Calabi-Yau manifolds |
| topic | Differential Geometry High Energy Physics - Theory 32Q25 |
| url | https://arxiv.org/abs/2405.19402 |