Numerical Hermitian Yang-Mills Connections and Kahler Cone Substructure

TL;DR

This paper advances numerical methods for computing Hermitian Yang-Mills connections on vector bundles over Calabi-Yau manifolds, enabling stability analysis across complex Kahler cone substructures with improved accuracy and efficiency.

Contribution

It extends the generalized Donaldson algorithm to bundles with Kahler cone substructure on manifolds with h^{1,1}>1, introducing normalized error measures, adaptive integration, and a stability check method.

Findings

Enhanced numerical algorithm for slope-stable bundles

Ability to probe stability across Kahler cone substructures

Improved integration and stability testing procedures

Abstract

We further develop the numerical algorithm for computing the gauge connection of slope-stable holomorphic vector bundles on Calabi-Yau manifolds. In particular, recent work on the generalized Donaldson algorithm is extended to bundles with Kahler cone substructure on manifolds with h^{1,1}>1. Since the computation depends only on a one-dimensional ray in the Kahler moduli space, it can probe slope-stability regardless of the size of h^{1,1}. Suitably normalized error measures are introduced to quantitatively compare results for different directions in Kahler moduli space. A significantly improved numerical integration procedure based on adaptive refinements is described and implemented. Finally, an efficient numerical check is proposed for determining whether or not a vector bundle is slope-stable without computing its full connection.