  We study the fate of reparametrization invariance of Wilson loops, also known
as 'zig-zag' symmetry, under the RG flow using some simple cases as guidance.
We restrict our analysis to large-$N$, strongly coupled CFTs and use the
holographic dual description of a Wilson loop as a fundamental string embedded
in asymptotically AdS spaces, at zero and nonzero temperature. We then
introduce a cutoff in the holographic radial direction and integrate out the
the section of the string closer to the AdS boundary in the spirit of
holographic Wilsonian renormalization. We make explicit the map between Wilson
loop reparametrizations and conformal transformation of the string worldsheet
and show that a cutoff anchored to the worldsheet breaks conformal invariance
and induces an effective defect action for reparametrizations at the cutoff
scale, in a way similar to nearly-$AdS_2$ gravity or SYK models. On the other
hand, a cutoff in the target space breaks worldsheet diffeomorphisms and Weyl
transformations but keeps conformal transformations unbroken and does not
generate a non-trivial action for reparametrizations.
