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I have a very basic question about Wilson lines (WL).

This is what I know about the WL:

  1. WL help us to learn about the important properties of gauge fields (treated as connections on the space of quantum fields). We can study parallel transport, holonomies, etc.

  2. WL are important in the studies of the (de-)confinement. Whether the loops 'want' to contract or not is in some relation with the binding potential; $\operatorname{e}^V\propto\oint(\text{Wilson Loop})$.

Now, the question:

In which theories do we normally consider such observables as WL? If, suppose, the WL can be considered in any theory containing gauge fields, then why do the standard texts on QED not talk about WL? On the other hand, if the WL are of interest in non-abelian theories only (where gluon-gluon interaction is present), then why are people not talking about WL when, say, considering gauged non-linear sigma models, or gravity, or the electroweak theory?

Many thanks!

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Wilson loops are mainly relevant in nonabelian Yang-Mills theories (e.g. QCD) where there is a transition from confined phase at low energies (with quarks and gluons bound together) to an asymptotically free one (where they can move around freely in a quark-gluon-plasma) at high energies. The reason is, as you roughly mentioned in your second point, that the Wilson loops can be used as an order parameter to distinguish between these two phases. For the electroweak case (which is also a nonabelian Yang-Mills theory) confinement is prevented as Higgs mechanism breaks the nonabelian gauge group of the theory at low energies. Therefore Wilson loops don't play an equally important role in the electroweak sector.

Wilson loops play therefore an important role, especially in the field of lattice QCD. One of their useful properties is that every local operator can be written in terms of Wilson loops. $F_{\mu\nu}(x)$ can for example be written as an infinitesimal Wilson loop around the point $x$. This allows people working in lattice QCD to rewrite lagrangians and operators as series of Wilson lines on a lattice.

Wilson loops are furthermore important when people are studying S-duality of the Yang-Mills equations. Under S-duality the (color-)electric and (color-)magnetic fields are interchanged, (color)charges are mapped to monopoles and the Wilson loops are mapped to 't Hooft lines.

Besides the normal Maxwell term $\text{Tr}[F_{\mu\nu} F^{\mu\nu}]$ in the action of nonabelian gauge theories (where $F_{\mu\nu} = D_\mu A_\nu - D_\nu A_\mu$), one can also consider a Chern-Simons term in the action of a nonabelian gauge theory (in an odd number of dimensions) $\text{Tr}[\epsilon^{\mu\nu\rho}A_\mu D_{\nu} A_\rho]$.

Though Wilson loops are sporadically used in other quantum field theories, their main application lies in nonabelian Yang-Mills (and Chern-Simons) gauge theories (and their supersymmetric extensions) due to their usefulness in studying the above phenomena.

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