2026/07/15 by Hanwen Liu
Mathematics · #math.DG #math-ph #math.MP
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 field h of positive definite symmetric matrices. We show that h determines a unique compatible connection A(h) and that the Yang--Mills equation is equivalent to the exactly determined second order system Φg(h):=FA(h)+h-1-g=0. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator. For matrix fields of the form h=e2ωg, the equation Φg(h)=0 is equivalent to anti-self-duality and constant scalar curvature 6√(2). Consequently, every anti-self-dual conformal class of positive Yamabe invariant gives a global solution.