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Numerical Hermitian Yang-Mills Connections and Vector Bundle Stability in Heterotic Theories

Lara B. Anderson, Volker Braun, Robert L. Karp, Burt A. Ovrut

TL;DR

This work develops and implements a generalized Donaldson algorithm to compute Hermitian Yang–Mills connections on slope-stable holomorphic vector bundles over Calabi–Yau manifolds, enabling direct numerical testing of bundle stability and supersymmetry conditions in heterotic vacua. By embedding twisted bundles into Grassmannians and using an extended T-operator, the authors obtain Hermitian–Einstein metrics in the large-k limit for slope-stable cases and extract the SU(n) gauge connection by untwisting via determinant line bundles. They introduce rigorous error measures for both Calabi–Yau metrics and bundle connections, demonstrate convergence on the Quartic K3 and Quintic, and show how the algorithm reflects Harder–Narasimhan filtrations for semi-stable and unstable bundles, including curvature singularities. The results provide a practical numerical tool to classify stability and to explore moduli-dependent phenomenology in heterotic compactifications, with potential applications to Yukawa couplings and stability-wall phenomena. Overall, the paper establishes a concrete pathway to compute key geometric data (metrics and connections) needed to quantitatively analyze supersymmetric vacua in string theory.

Abstract

A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.

Numerical Hermitian Yang-Mills Connections and Vector Bundle Stability in Heterotic Theories

TL;DR

This work develops and implements a generalized Donaldson algorithm to compute Hermitian Yang–Mills connections on slope-stable holomorphic vector bundles over Calabi–Yau manifolds, enabling direct numerical testing of bundle stability and supersymmetry conditions in heterotic vacua. By embedding twisted bundles into Grassmannians and using an extended T-operator, the authors obtain Hermitian–Einstein metrics in the large-k limit for slope-stable cases and extract the SU(n) gauge connection by untwisting via determinant line bundles. They introduce rigorous error measures for both Calabi–Yau metrics and bundle connections, demonstrate convergence on the Quartic K3 and Quintic, and show how the algorithm reflects Harder–Narasimhan filtrations for semi-stable and unstable bundles, including curvature singularities. The results provide a practical numerical tool to classify stability and to explore moduli-dependent phenomenology in heterotic compactifications, with potential applications to Yukawa couplings and stability-wall phenomena. Overall, the paper establishes a concrete pathway to compute key geometric data (metrics and connections) needed to quantitatively analyze supersymmetric vacua in string theory.

Abstract

A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.

Paper Structure

This paper contains 31 sections, 5 theorems, 93 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Generalized Donaldson Algorithm and Heterotic Compactifications
  3. The Hermitian Yang-Mills Equations and Hermitian-Ein-stein Bundle Metrics
  4. Slope Stability
  5. Computing the Calabi-Yau Metric
  6. Donaldson's Algorithm
  7. Ricci Curvature and Error Measures
  8. The Quartic $K3$
  9. The Quintic Threefold
  10. Hermitian Yang-Mills Connections
  11. Generalizing Donaldson's Algorithm
  12. Untwisting
  13. Measuring the Error
  14. An Example
  15. Subtracting the Trace
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\{s_\alpha \}$ be a basis for $H^0(X,\mathscr{L}^k)$ for some ample line bundle $\mathscr{L}$. Then the space of all "algebraic" Kähler potentials, where $k\in\mathbb{Z}$, is dense in the space of Kähler potentials.

Figures (15)

  • Figure 1: The error measures $\sigma_k$ defined in Subsection \ref{['sec:ricci']}. The data shown is for the Quartic $K3$ defined as a hypersurface in $\mathbb{P}^3$\ref{['k3']}. The complex structure parameter is chosen to be $\psi=\frac{1}{2}$ and $\psi=\frac{i}{2}$. Shown is data generated using the code developed in Braun:2007snBraun:2008jp (Code1) and data generated by the implementation in Douglas:2006rrDouglas:2006hz (Code2). The error measure is fitted to the theoretical convergence given in \ref{['sigma_decay']}.
  • Figure 2: Average scalar curvature for the metric on $K3 \subset \mathop{ {\mathbb{P}}}\nolimits^3$.
  • Figure 3: Curvature measures for $K3 \subset \mathop{ {\mathbb{P}}}\nolimits^3$. These provide the additional checks of numeric accuracy described in \ref{['sec:ricci']}.
  • Figure 4: The $\sigma_k$ error measures for the quintic threefold \ref{['quintic']} in $\mathbb{P}^4$. Shown is the error measure described in Subsection \ref{['sec:ricci']}, evaluated for the two values $\psi=\frac{1}{2}$ and $\psi=\frac{i}{2}$. Code1 and Code2 are associated to the implementations of Braun:2007snBraun:2008jp and Douglas:2006rrDouglas:2006hz respectively.
  • Figure 5: Average scalar curvature for the metric on the Quintic, \ref{['quintic']}.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Theorem 1: Tian
  • Theorem 2: Donaldson
  • Theorem 3: Donaldson
  • Theorem 4: Wang
  • Theorem 5: Harder-Narasimhan