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

2025-12-11

AI Summary

This work addresses the precise computation of physical Yukawa couplings in heterotic string compactifications. We present the first general numerical solver for Hermitian Yang–Mills (HYM) connections on holomorphic vector bundles of arbitrary rank and non-Abelian structure group over Calabi–Yau threefolds. Our method introduces a geometric machine learning framework intrinsically incorporating differential-geometric structures: it combines holomorphic section basis enumeration, alternating gradient optimization, and numerical connection evolution on the Kähler moduli space—requiring no symmetry assumptions or model simplifications. Crucially, this approach overcomes longstanding limitations of traditional HYM solvers, which rely on Abelian structure groups or highly symmetric geometries. We compute all physically normalized Yukawa couplings for compactification models with non-Abelian gauge bundles, achieving relative precision of $10^{-4}$. These results rigorously confirm both the existence and observational viability of supersymmetric vacua in realistic heterotic constructions.

Technology Category

• Machine Learning: Learning with Manifolds
• Humans and AI: Other Foundations of Human Computation & AI
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Computes Hermitian Yang-Mills connections on vector bundles using geometric machine learning
• Enables calculation of Yukawa couplings in heterotic string compactifications
• Applies method to bundles with non-Abelian structure groups in full computation

Innovation

• Alternating optimization procedure using geometric machine learning
• General method for any rank and structure group of bundles
• Computes Yukawa couplings in heterotic string compactifications