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

For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\mathrm{SO}(3)$ connections on the bundle ${\mathrm{\Lambda }}^{+}$ 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 $${\mathrm{\Phi }}_g(h):=F_{A(h)}^{+}{h}^{-1}-g=0.$$ We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator. For matrix fields of the form $h=e^{2\omega }g$, the equation ${\mathrm{\Phi }}_g(h)=0$ is equivalent to anti-self-duality and constant scalar curvature $6\sqrt{2}$. Consequently, every anti-self-dual conformal class of positive Yamabe invariant gives a global solution.