تقرير
Harmonic $1$-forms on real loci of Calabi-Yau manifolds
| العنوان: | Harmonic $1$-forms on real loci of Calabi-Yau manifolds |
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| المؤلفون: | Douglas, Michael R., Platt, Daniel, Qi, Yidi |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics High Energy Physics - Theory |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, High Energy Physics - Theory, 32Q25 |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.19402 |
| رقم الأكسشن: | edsarx.2405.19402 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |