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Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds

Elli Heyes, Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva

Abstract

A numerical framework for approximating $\mathrm{G}_2$-structure 3-forms on contact Calabi-Yau manifolds is presented. The approach proceeds in three stages: first, existing neural network models are employed to compute an approximate Ricci-flat metric on a Calabi-Yau threefold. Second, using this metric and the explicit construction of a $\mathrm{G}_2$-structure on the associated 7-dimensional Calabi-Yau link in the 9-sphere, numerical approximations of the 3-form are generated on a large set of sampled points. Finally, a dedicated neural architecture is trained to learn the 3-form and its induced Riemannian metric directly from data, validating the learned structure and its torsion via a numerical implementation of the exterior derivative, which may be of independent interest.

Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds

Abstract

A numerical framework for approximating -structure 3-forms on contact Calabi-Yau manifolds is presented. The approach proceeds in three stages: first, existing neural network models are employed to compute an approximate Ricci-flat metric on a Calabi-Yau threefold. Second, using this metric and the explicit construction of a -structure on the associated 7-dimensional Calabi-Yau link in the 9-sphere, numerical approximations of the 3-form are generated on a large set of sampled points. Finally, a dedicated neural architecture is trained to learn the 3-form and its induced Riemannian metric directly from data, validating the learned structure and its torsion via a numerical implementation of the exterior derivative, which may be of independent interest.
Paper Structure (20 sections, 4 theorems, 35 equations, 9 figures)

This paper contains 20 sections, 4 theorems, 35 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Key concepts in special geometries
  4. Outline
  5. Background
  6. G2-geometry
  7. Contact Calabi--Yau manifolds
  8. Calabi--Yau links
  9. Data Generation & Analysis
  10. Generating the base CY metric
  11. Numerical checks on cymetric package
  12. Constructing G2-structures numerically
  13. Analysis of the numerical G2 dataset
  14. G2-Structure Learning & Analysis
  15. Neural architecture
  16. ...and 5 more sections

Key Result

Proposition 2

Let $\varphi \in \Omega^{3}(M)$, and let $x^{1},\ldots,x^{7}$ be local coordinates on an open set $U \subset M$. For $i,j \in \{1,\ldots,7\}$, define local smooth functions $B_{ij}$ by If $\varphi$ defines a ${\rm G}_{2}$-structure with associated Riemannian metric $g_\varphi$, then Consequently, and the metric components are recovered from $\varphi$ by

Figures (9)

  • Figure 1: Variation of the Kähler defect with sampling radius $\varepsilon$, via \ref{['eq:medianNED']}.
  • Figure 2: Pointwise values of $\frac{\varphi\wedge\psi}{\mathrm{Vol}_{g_\varphi}}$ across the dataset.
  • Figure 3: Comparison of the volume densities $\sqrt{\mathop\mathrm{det}\nolimits g_{\varphi}}$ and $\sqrt{\mathop\mathrm{det}\nolimits g_{\mathrm{CY}}}$.
  • Figure 4: Histogram of the 35 independent components of the $\mathrm{G}_2$ three-form $\varphi$.
  • Figure 5: Histogram of the 28 independent components of the $\mathrm{G}_2$ metric $g_{\varphi}$.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Definition 1
  • Proposition 2: Karigiannis2008karigiannis2008notesg2spin7geometry
  • Theorem 3: Fernandez1982
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7: Habib2015Calvo-Andrade:2016ftiGray1969
  • Definition 8
  • Proposition 9: Habib2015