Machine Learned Calabi-Yau Metrics and Curvature
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911908842438656 |
|---|---|
| author | Berglund, Per Butbaia, Giorgi Hübsch, Tristan Jejjala, Vishnu Peña, Damián Mayorga Mishra, Challenger Tan, Justin |
| author_facet | Berglund, Per Butbaia, Giorgi Hübsch, Tristan Jejjala, Vishnu Peña, Damián Mayorga Mishra, Challenger Tan, Justin |
| contents | Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $χ(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09801 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Machine Learned Calabi-Yau Metrics and Curvature Berglund, Per Butbaia, Giorgi Hübsch, Tristan Jejjala, Vishnu Peña, Damián Mayorga Mishra, Challenger Tan, Justin High Energy Physics - Theory Machine Learning Algebraic Geometry Differential Geometry Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $χ(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations. |
| title | Machine Learned Calabi-Yau Metrics and Curvature |
| topic | High Energy Physics - Theory Machine Learning Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2211.09801 |