Wilson loop

Remarks

The value does not depend on $p$: another point $p g$ of the fibre gives the holonomy $g^{-1} \mathrm{Hol}_A(\gamma, p) \, g$, and the trace is invariant under conjugation. For the same reason $W_\rho(\gamma, A)$ is unchanged when $A$ is moved by an element of the gauge group. On a trivial bundle with a matrix group, physicists write $W_\rho(\gamma, A) = \operatorname{tr} \mathcal{P} \exp\left( \oint_\gamma A \right)$, a path-ordered exponential, with signs and factors of $i$ that depend on conventions.

Example: $G = U(1)$, $\rho$ the identity representation, $M = \R^2$, and $A = i\alpha$ for a real $1$-form $\alpha$. Then the holonomy commutes with everything, $W(\gamma, A) = \exp\left(\pm i \oint_\gamma \alpha\right)$ with the sign fixed by the convention, and by Stokes' theorem $\oint_\gamma \alpha = \int_D d\alpha$ when $\gamma$ bounds a disc $D$. For $\alpha = \frac{B}{2}(x \, dy - y \, dx)$ and $\gamma$ the circle of radius $r$, this gives $W = e^{\pm i B \pi r^2}$, the exponential of the flux.

Non-example: for a non-abelian group, the holonomy $\mathrm{Hol}_A(\gamma, p)$ itself is not a Wilson loop and is not gauge invariant. It changes by conjugation, and only class functions of it, such as traces, are invariant.

In a quantum gauge theory, or in lattice gauge theory, one studies the expectation value $\langle W_\rho(\gamma, A) \rangle$. For a rectangle of sides $R$ and $T$ in the time direction, decay like $e^{-\sigma R T}$ with $\sigma > 0$, the area law, is Wilson's criterion for confinement.