Yang-Mills vacuum ($F_{ij}=0$)

A classical zero-energy Yang-Mills vacuum has vanishing spatial gauge curvature. On simply connected space it is pure gauge, $A=h^{-1}dh$. If gauge transformations approach the identity at spatial infinity, the boundary-normalized gauge maps on compactified space $S^3$ have integer winding, recorded by the Chern-Simons number of a gauge field. Transformations of nonzero winding relate these representatives; quantum theta states encode their tunnelling and large-gauge-transformation phases.