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Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

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

TL;DR

We develop and test a numerical framework to solve the hermitian Yang-Mills equations on stable holomorphic vector bundles by extending Donaldson's balanced-metric method, including a generalized T-operator. The method uses projective embeddings and the density-of-states expansion to approximate Hermitian-Einstein metrics, with applications to the tangent bundle of $P^n$, a stable rank-3 bundle on $P^2$, and a rank-3 bundle on the Fermat quintic, demonstrating convergence of balanced metrics and approximate satisfaction of the Hermitian-Yang-Mills equations as well as Ricci-flat metrics in Calabi-Yau settings. These results provide a practical computational toolkit for accessing metrics and connections needed in string compactifications where the Kahler potential and Yukawa couplings depend on the balanced/Hermitian-Einstein data. The work lays groundwork for computing EFT-relevant quantities by numerically accessing Ricci-flat and Hermitian-Yang-Mills structures on Calabi-Yau manifolds and their bundles.

Abstract

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.

Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

TL;DR

We develop and test a numerical framework to solve the hermitian Yang-Mills equations on stable holomorphic vector bundles by extending Donaldson's balanced-metric method, including a generalized T-operator. The method uses projective embeddings and the density-of-states expansion to approximate Hermitian-Einstein metrics, with applications to the tangent bundle of , a stable rank-3 bundle on , and a rank-3 bundle on the Fermat quintic, demonstrating convergence of balanced metrics and approximate satisfaction of the Hermitian-Yang-Mills equations as well as Ricci-flat metrics in Calabi-Yau settings. These results provide a practical computational toolkit for accessing metrics and connections needed in string compactifications where the Kahler potential and Yukawa couplings depend on the balanced/Hermitian-Einstein data. The work lays groundwork for computing EFT-relevant quantities by numerically accessing Ricci-flat and Hermitian-Yang-Mills structures on Calabi-Yau manifolds and their bundles.

Abstract

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.

Paper Structure

This paper contains 16 sections, 8 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Metrics of constant scalar curvature
  3. Approximating Ricci-flat metrics by projective embedding
  4. Balanced metrics
  5. Finding the balanced metric
  6. Balanced metrics and constant scalar curvature
  7. The hermitian Yang-Mills equations
  8. Embedding vector bundles
  9. Generalized T-operator
  10. Generalized T-operator
  11. Examples
  12. Hermite-Einstein metric on the tangent bundle of $\mathbb{P}^n$
  13. A stable rank 3 bundle over $\mathbb{P}^2$
  14. A rank 3 bundle on the Fermat quintic
  15. Ricci flat metric on Fermat quintic
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose the automorphism group ${\operatorname{Aut}}(X,{\mathcal{L}})$ is discrete. If $(X,{\mathcal{L}}^k)$ is balanced, then the choice of basis in $H^0(X,{\mathcal{L}}^k)$ such that $i_k({\mathcal{L}})$ is balanced is unique up to the action of $U(N)\times \mathbb{R}^*$.

Figures (2)

  • Figure 1: The shape of the rational curve for the balanced and non-balanced metrics.
  • Figure 2: The probability density in the radial direction on the rational curve.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Conjecture 3.5