Pure gauge potential ($A=-dg{g}^{-1}$)

With gauge covariant derivative $D=d+A$, transforming the zero connection by a smooth group-valued map $g$ gives $A=-dg{g}^{-1}$. This potential has zero gauge curvature. A flat potential is locally of this form; global reconstruction can also depend on topology and holonomy. The sign changes when the definition of the covariant derivative changes.