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

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 $S^4$.

💡 Research Summary

The paper addresses Yau’s Problem 117, which asks whether every Yang‑Mills connection on the four‑sphere S⁴ must be either self‑dual (an instanton) or anti‑self‑dual. Earlier work proved this under restrictive hypotheses: weak stability together with specific structure groups such as SU(2), SU(3) or U(2). For SO(4) and higher‑rank simple groups, non‑self‑dual Yang‑Mills fields are known to exist, but they are either unstable or fail to be irreducible. The present work removes the group‑specific restrictions by imposing two natural conditions: (i) the Lie algebra 𝔤 of the structure group G is simple, and (ii) the connection is irreducible, i.e., its holonomy equals the whole group G.

The authors first recall the Yang‑Mills functional YM(D)=½‖F_D‖², the Euler‑Lagrange equation d*_D F_D=0, and the second variation operator (the Jacobi operator)
F_D(B)=Δ_D B+R_D(B)=D*D B+ B∘Ric+2 R_D(B).
A connection is called weakly stable if ⟨F_D B,B⟩≥0 for all 𝔤‑valued 1‑forms B. On an oriented four‑manifold the Hodge star satisfies *²=Id on 2‑forms, so the curvature splits as F_D=F⁺+F⁻ with *F⁺=+F⁺ and *F⁻=−F⁻.

A key algebraic observation (Lemma 2.3, originally due to Stern) is that for a weakly stable Yang‑Mills connection on S⁴ the sub‑algebras generated by the self‑dual and anti‑self‑dual parts, denoted K⁺_x and K⁻_x, commute pointwise: