A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory

Abstract

For a closed oriented Riemannian $4$-manifold (M,g), we consider SO(3) connections on the bundle Λ⁺ of self-dual $2$-forms. On the open locus where the self-dual curvature is an orientation-preserving frame, pointwise polar decomposition removes the gauge freedom and replaces the connection by a positive self-adjoint endomorphism field h of Λ⁺. Using the classical reconstruction of the compatible connection A(h), we obtain a global formulation of the Yang--Mills equation as the determined second order system Φ_g(h):=F_A(h)⁺h⁻¹−Id=0. Here, the tensor F_A(h)⁺ is the self-dual curvature of A(h), regarded as an endomorphism of Λ⁺, and Id is the identity of Λ⁺. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator D_hΦ_g. For fields of the form h=e^2ωId, where ω is a smooth real-valued function on M, the equation Φ_g(h)=0 is equivalent to anti-self-duality and constant scalar curvature 6√2 of the conformal metric ĝ=e^2ωg. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of Φ_g(h)=0.

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BibTeX RIS

BibTeX

@misc{liu2026selfdual,
  title = {{A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory}},
  author = {Hanwen Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.14204},
  eprint = {2607.14204},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.14204},
}

RIS

TY  - UNPB
TI  - A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory
AU  - Hanwen Liu
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2607.14204
UR  - https://arxiv.org/abs/2607.14204
AB  - For a closed oriented Riemannian $4$-manifold (M,g), we consider SO(3) connections on the bundle Λ^+ of self-dual $2$-forms. On the open locus where the self-dual curvature is an orientation-preserving frame, pointwise polar decomposition removes the gauge freedom and replaces the connection by a positive self-adjoint endomorphism field h of Λ^+. Using the classical reconstruction of the compatible connection A(h), we obtain a global formulation of the Yang--Mills equation as the determined second order system Φ_g(h):=F_A(h)^+h^−1−Id=0. Here, the tensor F_A(h)^+ is the self-dual curvature of A(h), regarded as an endomorphism of Λ^+, and Id is the identity of Λ^+. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator D_hΦ_g. For fields of the form h=e^2ωId, where ω is a smooth real-valued function on M, the equation Φ_g(h)=0 is equivalent to anti-self-duality and constant scalar curvature 6√2 of the conformal metric ĝ=e^2ωg. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of Φ_g(h)=0.
ER  -

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