# Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

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.)

• Recall that the matrix-valued Wilson loop in (non-abelian) lattice QFT is not gauge invariant, so calling it "a conserved quantity" seems suspect. (Its trace is the gauge-invariant object.) The form $M = I + igA_{\alpha} \delta x^{\alpha} + \dots$ is needed in order for typical lattice actions based on Wilson loops to correctly reproduce tr$F_{\mu\nu} F^{\mu\nu}$ in the continuum limit. It might be possible to design alternative lattice actions that don't require links in this form, but I'm not aware of anything like that. Your 2-form looks abelian; should it have a wedge-product term? – David Schaich Dec 15 '18 at 13:48