{
  "abstract": "For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\\operatorname{SO}(3)$ 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 $$ Φ_g(h):=F_{A(h)}^+h^{-1}-\\operatorname{Id}=0. $$ Here, the tensor $F_{A(h)}^+$ is the self-dual curvature of $A(h)$, regarded as an endomorphism of $Λ^+$, and $\\operatorname{Id}$ 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 $h=e^{2ω}\\operatorname{Id}$, 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 $6\\sqrt{2}$ of the conformal metric $\\hat g=e^{2ω}g$. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of $Φ_g(h)=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14204",
  "author_total": 1,
  "authors": [
    {
      "name": "Hanwen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7j41qnhnv3v89.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14204",
  "fulltext_url": null,
  "id": "7j41qnhnv3v89",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14204",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7169580217",
  "pdf_url": "https://arxiv.org/pdf/2607.14204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7j41qnhnv3v89.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2607.14204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory",
  "title_text": "A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/7j41qnhnv3v89",
  "venue": "arXiv",
  "versions": []
}