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Numerical Calabi-Yau metrics

Michael R. Douglas, Robert L. Karp, Sergio Lukic, Rene Reinbacher

TL;DR

The paper develops numerical methods to approximate Ricci-flat metrics on Calabi–Yau hypersurfaces via balanced metrics from Kodaira embeddings and the T-map, with convergence guaranteed under discrete automorphism groups. It introduces a Monte Carlo-based numerical integration framework for integrals on CY varieties using the volume form $d\mu_\Omega=\Omega\wedge\overline{\Omega}$ and exploits mass functions and symmetry reductions to stabilize and accelerate computation, demonstrated on a one-parameter family of quintics in ${\mathbb P}^4$. The results show that the balanced metrics converge to the Ricci-flat metric as $k$ increases, with the deviation measure $\sigma_k$ shrinking roughly as $\alpha/k^2+\beta/k^3$ and Ricci scalars decreasing accordingly. The discussion covers efficiency, error control, and possible extensions to Hermitian Yang–Mills connections and more general Calabi–Yau constructions.

Abstract

We develop numerical methods for approximating Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics, and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.

Numerical Calabi-Yau metrics

TL;DR

The paper develops numerical methods to approximate Ricci-flat metrics on Calabi–Yau hypersurfaces via balanced metrics from Kodaira embeddings and the T-map, with convergence guaranteed under discrete automorphism groups. It introduces a Monte Carlo-based numerical integration framework for integrals on CY varieties using the volume form and exploits mass functions and symmetry reductions to stabilize and accelerate computation, demonstrated on a one-parameter family of quintics in . The results show that the balanced metrics converge to the Ricci-flat metric as increases, with the deviation measure shrinking roughly as and Ricci scalars decreasing accordingly. The discussion covers efficiency, error control, and possible extensions to Hermitian Yang–Mills connections and more general Calabi–Yau constructions.

Abstract

We develop numerical methods for approximating Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics, and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.

Paper Structure

This paper contains 14 sections, 4 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. T-map
  3. Numerical integration on Calabi-Yau varieties
  4. Basic setup
  5. Generating the sample
  6. Expected values of currents
  7. The Calabi-Yau case
  8. The expected zero current
  9. The numerical mass
  10. Symmetries
  11. Numerical results
  12. Approximating volumes v.s. Calabi-Yau volume
  13. Ricci scalars
  14. Discussion

Key Result

Theorem 2.1

(see, e.g. ,SanoDonaldson1; and Bourguignon:YauDonaldson:numeric for the $\nu$-balanced case) Suppose that ${\operatorname{Aut}}(X,L)$ is discrete. If a balanced embedding exists, then, for any initial hermitian matrix $G_0$, as $r\to \infty$ the sequence $T^r(G_0)$ converges to a fixed point.

Figures (3)

  • Figure 1: $\sigma_k$ and Ricci scalars.
  • Figure 2: The values of $\eta$ on the rational curve, for $k=1$, 3, 4, 5, 7, 9, 11 and 12.
  • Figure 3: Histogram of Ricci scalars, and scattered plot of mass versus the Ricci scalars.

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2