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Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds

Daattavya Aggarwal, Yang-Hui He, Elli Heyes, Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva

TL;DR

Comparing datasets for certain Sasakian Hodge numbers and the Crowley-Nordstrom invariant of the natural $G_2$-structure of the natural $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity leads to a vast improvement in computation speeds which may be of independent interest.

Abstract

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.

Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds

TL;DR

Comparing datasets for certain Sasakian Hodge numbers and the Crowley-Nordstrom invariant of the natural -structure of the natural -dimensional link of a weighted projective Calabi-Yau -fold hypersurface singularity leads to a vast improvement in computation speeds which may be of independent interest.

Abstract

We propose a machine learning approach to study topological quantities related to the Sasakian and -geometries of contact Calabi-Yau -manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-Nördstrom invariant of the natural -structure of the -dimensional link of a weighted projective Calabi-Yau -fold hypersurface singularity, for 7549 of the 7555 possible projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the weights alone, using both neural networks and a symbolic regressor which achieve 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.
Paper Structure (21 sections, 3 theorems, 43 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 3 theorems, 43 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Topological invariants of CY links
  4. Machine Learning cCY topology
  5. Background
  6. Calabi-Yau links
  7. Sasakian Hodge numbers of a CY link
  8. The Crowley-Nordström invariant on cCY $7$-manifolds
  9. Weak R-equivalence of weighted polynomials
  10. Machine Learning
  11. Data Generation & Analysis
  12. Sasakian Hodge Numbers
  13. Crowley-Nördtrom invariant
  14. Explicit Weak R-Equivalence
  15. Machine Learning
  16. ...and 6 more sections

Key Result

Theorem 2

Let $f$ be a ${\bf w}$-homogeneous polynomial on $\mathbb{C}^n$ of degree $d$. Given $p+q=n$, let $\ell=(p+1)d-\sum_i w_i$, and denote by $(\mathbb{M}_f)_\ell$ the linear subspace of the Milnor algebra consisting of degree $\ell$ elements. When CYcondition is satisfied, i.e. $K_f$ is a Calabi-Yau link, the condition reduces to $\ell = pd$.

Figures (5)

  • Figure 1: Histogram of Gröbner basis lengths for the 7549 Calabi-Yau link constructions computed.
  • Figure 2: Histogram of Sasakian $h^{2,1}$ values for the 7549 Calabi-Yau link constructions computed.
  • Figure 3: Scatter graph of the Calabi-Yau complex threefold $h^{2,1}$ values against the Sasakian transverse $h^{2,1}$ values for the 7549 Calabi-Yau link constructions considered. For this data Sasakian $h^{2,1} \leq$ CY $h^{2,1}$, with 4198 cases satisfying the equality.
  • Figure 4: Histogram of CN invariants for the 7549 Calabi-Yau link constructions computed.
  • Figure 5: ML architecture predictions of the $h_S^{2,1}$ values, against the true values, for the 7549 Calabi-Yau link constructions considered, from: (a) a trained NN; and (b) the symbolic regression best model of equation \ref{['eq:SymReg_eq']}.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 2: Itoh2004, Steenbrink1977Steenbrink1983
  • Proposition 3: Habib2015
  • Definition 4
  • Theorem 5
  • Definition 6
  • Conjecture 7
  • Conjecture 8