Deviation from Alday-Maldacena Duality For Wavy Circle

TL;DR

This paper investigates deviations from the Alday-Maldacena duality for smooth wavy circle boundaries, correcting previous calculations and explicitly evaluating the first deviation terms in the minimal surface area and Wilson loop integral.

Contribution

It corrects a mistake in prior work and provides explicit calculations of the initial deviation terms for the wavy circle case, enhancing understanding of the duality's limitations.

Findings

Deviation from duality appears at second order in perturbation.

Explicit expressions for deviation terms are derived.

The correction clarifies previous computational inaccuracies.

Abstract

Alday-Maldacena conjecture is that the area A_P of the minimal surface in AdS_5 space with a boundary P, located in Euclidean space at infinity of AdS_5, coincides with a double integral D_P along P, the Abelian Wilson average in an auxiliary dual model. The boundary P is a polygon formed by momenta of n external light-like particles in N=4 SYM theory, and in a certain n=infty limit it can be substituted by an arbitrary smooth curve (wavy circle). The Alday-Maldacena conjecture is known to be violated for n>5, when it fails to be supported by the peculiar global conformal invariance, however, the structure of deviations remains obscure. The case of wavy lines can appear more convenient for analysis of these deviations due to the systematic method developed in arXiv:0803.1547 for (perturbative) evaluation of minimal areas, which is not yet available in the presence of angles at finite n. We correct a mistake in that paper and explicitly evaluate the h^2\bar h^2 terms, where the first deviation from the Alday-Maldacena duality arises for the wavy circle.