Deviation from Alday-Maldacena duality for wavy circle

Alday-Maldacena conjecture is stated in this paper that the area AΠ of the minimal surface in AdS5 space with a boundary Π, located in Euclidean space at infinity of AdS5, coincides with a double integral DΠ along Π, the Abelian Wilson average in an auxiliary dual model. This comes from Alday and Maldacena's original proposal and the BDS conjecture on the extrapolation of the MHV amplitudes. The boundary Π is a polygon formed by momenta of n external light-like particles in N=4 SYM theory, and in a certain n=∞ 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 dual 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 [H. Itoyama, A. Mironov, A. Morozov, Anomaly in n=∞ Alday-Maldacena duality for wavy circle, JHEP 0807 (2008) 024, 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 h2h¯2 terms, where the first deviation from the Alday-Maldacena duality arises for the wavy circle.