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Weakly stable irreducible Yang-Mills fields over $S^4$

Jianquan Ge, Lixin Xiao

Abstract

Addressing Yau's conjecture (Problem 117) on $S^4$, we investigate the self-duality of weakly stable Yang-Mills fields under the assumption of irreducibility. For structure groups with a simple Lie algebra, we prove that any weakly stable irreducible connection must be either self-dual or anti-self-dual. Furthermore, we demonstrate that if the Lie algebra admits a non-trivial abelian center, no irreducible Yang-Mills fields can exist over $S^4$.

Weakly stable irreducible Yang-Mills fields over $S^4$

Abstract

Addressing Yau's conjecture (Problem 117) on , we investigate the self-duality of weakly stable Yang-Mills fields under the assumption of irreducibility. For structure groups with a simple Lie algebra, we prove that any weakly stable irreducible connection must be either self-dual or anti-self-dual. Furthermore, we demonstrate that if the Lie algebra admits a non-trivial abelian center, no irreducible Yang-Mills fields can exist over .
Paper Structure (3 sections, 5 theorems, 12 equations)

This paper contains 3 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Theorem

Key Result

Theorem 1.2

BL81 Any weakly stable Yang-Mills field on $S^{4}$ with structure group $G=SU(2), SU(3)$ or $U(2)$ is either self-dual or anti-self-dual.

Theorems & Definitions (8)

  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 2.1: Ambrose-Singer AS53
  • Definition 2.2
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['thm-main']}
  • proof : Proof of Proposition \ref{['no irreducible']}