Hermitian Yang--Mills connections on general vector bundles: geometry and physical Yukawa couplings

TL;DR

This paper introduces a geometric machine learning method to compute Hermitian Yang-Mills connections on vector bundles, enabling the calculation of physical Yukawa couplings in heterotic string compactifications with complex gauge bundles.

Contribution

It presents a fully general alternating optimisation procedure based on geometric machine learning for solving Hermitian Yang-Mills equations on vector bundles of any rank and structure group.

Findings

Successfully computed Yukawa couplings for a heterotic compactification with non-Abelian gauge bundle.

Demonstrated the method's applicability to a broad class of vector bundles.

Provided a new computational approach for string theory compactification models.

Abstract

We compute solutions to the Hermitian Yang-Mills equations on holomorphic vector bundles $V$ via an alternating optimisation procedure founded on geometric machine learning. The proposed method is fully general with respect to the rank and structure group of $V$, requiring only the ability to enumerate a basis of global sections for a given bundle. This enables us to compute the physically normalised Yukawa couplings in a broad class of heterotic string compactifications. Using this method, we carry out this computation in full for a heterotic compactification incorporating a gauge bundle with non-Abelian structure group.