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

TL;DR

This paper introduces a numerical method to compute gauge connections on vector bundles over Calabi-Yau manifolds, enabling the assessment of stability and solutions to Hermitian Yang-Mills equations crucial for heterotic string theories.

Contribution

The authors develop a novel numerical algorithm for explicitly calculating gauge connections on holomorphic vector bundles, including stability analysis and approximation of Hermitian Yang-Mills solutions.

Findings

Successfully computed connections on stable monad bundles on K3 and Quintic threefolds.

Introduced an error measure to evaluate the approximation quality of the Hermitian Yang-Mills solutions.

Demonstrated the ability to numerically distinguish stable from non-stable bundles, including singular cases.

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.