Holographic RG flow and reparametrization invariance of Wilson Loops

TL;DR

This paper investigates how reparametrization invariance of Wilson loops behaves under RG flow in holographic duals of strongly coupled CFTs, revealing that boundary cutoffs break certain symmetries and induce defect actions.

Contribution

It demonstrates the effects of holographic radial cutoffs on Wilson loop symmetries and maps reparametrizations to conformal transformations of the string worldsheet.

Findings

Cutoffs at the worldsheet break conformal invariance and induce defect actions.

Target space cutoffs break worldsheet diffeomorphisms but preserve conformal invariance.

Reparametrization symmetry is sensitive to the type of cutoff applied in holographic models.

Abstract

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.