 Anti-self-dual connection

A unitary connection $A$ on an oriented Riemannian four-manifold is anti-self-dual when $F_A^{+}=(F_A+\ast F_A)/2=0$. Its curvature is covariantly closed by the Bianchi identity and solves the Yang-Mills equations. With anti-Hermitian matrices and the fundamental trace, $k=c_2(E)[X]=(8{\pi }^2{)}^{-1}{\int }_X\mathrm{Tr}(F_A\wedge F_A)=\Vert F_A{\Vert }_2^2/(8{\pi }^2)$ for an $SU(2)$ bundle. Thus its charge is nonnegative.

// Destination: geometry-and-topology.bigb