The path-ordered exponential from which the Wilson loop is traced is, crudely,
$$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$
which returns a matrix $\mathbf{W}$ in the Lie group in question. (This should be the Aharonov-Bohm phase for U(1).) Since this applies to closed loops, it equivalently maps surfaces to Lie group elements based only on their boundaries, and so $\mathbf{W}$ is a conserved quantity with a net flux of 0 through a closed surface (the limit of an infinitely small loop).
But this looks suspiciously like the derivation of the Faraday 2-form, where $ \mathbf{F} = \mathrm{d}\mathbf{A} $ (implying a loop-integral relationship between $\mathbf{A}$ and $\mathbf{F}$) and so $\mathrm{d}\mathbf{F}=\mathrm{d}^2\mathbf{A}=0$ (conservation of the loop flux), and thus $\mathrm{d}{\star}\mathbf{F}=\mathbf{J}$ (4-current is the dual of the loop flux) and $\mathrm{d}^2{\star}\mathbf{F}=\mathrm{d}\mathbf{J}=0$ (conservation of that dual).
Question: Is the POE value $\mathbf{W}$ the field value? Could it be treated as a 2-form? Does its dual correspond to a useful 4-current / Could you work out another conserved quantity (as $\mathrm{d}^2{\star}\mathbf{F}=\mathrm{d}\mathbf{J}=0$) from its dual?
Follow-up Question: In lattice QFT, do the links always have to have values $ M = I + i A_{\alpha} \delta x^\alpha$ (corresponding to the differential segments of the path-ordered exponential)?
Follow-up Question 2: What relationship is impled between 2 nodes separated by a path of links whose integral equals some Lie group element $g$? Say for SU(3) whose nodes are in $C^3$; are the 2 node values related by $g$?
(Plz don't bite the newcomer.)
