Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| author | Aggarwal, Daattavya He, Yang-Hui Heyes, Elli Hirst, Edward Earp, Henrique N. Sá Silva, Tomás S. R. |
| author_facet | Aggarwal, Daattavya He, Yang-Hui Heyes, Elli Hirst, Edward Earp, Henrique N. Sá Silva, Tomás S. R. |
| contents | We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-Nördstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gröbner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_03064 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds Aggarwal, Daattavya He, Yang-Hui Heyes, Elli Hirst, Edward Earp, Henrique N. Sá Silva, Tomás S. R. Differential Geometry High Energy Physics - Theory Algebraic Geometry We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-Nördstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gröbner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised. |
| title | Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds |
| topic | Differential Geometry High Energy Physics - Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2310.03064 |