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

Abstract

For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\operatorname{SO}(3)$ connections on the bundle $\Lambda^+$ 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 positive self-adjoint endomorphism field $h$ of $\Lambda^+$. 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 $$ \Phi_g(h):=F_{A(h)}^+h^{-1}-\operatorname{Id}=0. $$ Here, the tensor $F_{A(h)}^+$ is the self-dual curvature of $A(h)$, regarded as an endomorphism of $\Lambda^+$, and $\operatorname{Id}$ is the identity of $\Lambda^+$. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator $D_h\Phi_g$. For fields of the form $h=e^{2\omega}\operatorname{Id}$, where $\omega$ is a smooth real-valued function on $M$, the equation $\Phi_g(h)=0$ is equivalent to anti-self-duality and constant scalar curvature $6\sqrt{2}$ of the conformal metric $\hat g=e^{2\omega}g$. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of $\Phi_g(h)=0$.