Yang-Mills, Complex Structures and Chern's Last Theorem

TL;DR

This paper explores the connection between Chern's approach to the non-existence of complex structures on the six-sphere and Yang-Mills theories, proposing a broader framework linking geometry and gauge theories.

Contribution

It introduces a novel perspective connecting Chern's arguments with Yang-Mills theories, suggesting a general method to study manifold structures through gauge theory insights.

Findings

Chern's approach can be applied to analyze manifold complex structures.

Examples show the applicability of the method to various manifolds.

Potential implications for understanding geometric structures via gauge theories.

Abstract

Recently Shiing-Shen Chern suggested that the six dimensional sphere $\mathbb{S}^6$ has no complex structure. Here we explore the relations between his arguments and Yang-Mills theories. In particular, we propose that Chern's approach is widely applicable to investigate connections between the geometry of manifolds and the structure of gauge theories. We also discuss several examples of manifolds, both with and without a complex structure.