Wilson line

A Wilson line is the path-ordered parallel transporterUnder a gauge transformation $V$, it transforms as $U(y,x)\mapsto V(y)U(y,x)V(x{)}^{-1}$.

With gauge covariant derivative $D=d+A$, a Wilson loop is the matrix trace in a specified group representation of parallel transport around a closed curve. The holonomy of a connection changes by conjugation under gauge equivalence of principal connections, so its trace is gauge-invariant. It detects global connection data even when the local gauge curvature is zero.

Expanding a short Wilson line determines the inhomogeneous gauge-potential transformation. For $V=1+i\alpha$ and Hermitian generators,