"Anomaly" in n=infinity Alday-Maldacena Duality for Wavy Circle

TL;DR

This paper investigates the Alday-Maldacena duality for n=infinity polygons, focusing on continuous curves like circles, and finds that boundary regularization affects the minimal area, revealing a slight anomaly in the duality.

Contribution

It provides a detailed analysis of the duality for continuous boundary curves and uncovers how regularization procedures influence the minimal surface calculations.

Findings

Minimal area depends strongly on boundary regularization.

Without boundary terms, the area is regularization independent but differs from the double-loop integral.

The duality is slightly violated, indicating an anomaly.

Abstract

If the Alday-Maldacena version of string/gauge duality is formulated as an equivalence between double loop and area integrals a la arXiv: 0708.1625, then this pure geometric relation can be tested for various choices of n-side polygons. The simplest possibility arises at n=infinity, with polygon substituted by an arbitrary continuous curve. If the curve is a circle, the minimal surface problem is exactly solvable. If it infinitesimally deviates from a circle, then the duality relation can be studied by expanding in powers of a small parameter. In the first approximation the Nambu-Goto (NG) equations can be linearized, and the peculiar NG Laplacian plays the central role. Making use of explicit zero-modes of this operator (NG-harmonic functions), we investigate the geometric duality in the lowest orders for small deformations of arbitrary shape lying in the plane of the original circle. We find a surprisingly strong dependence of the minimal area on regularization procedure affecting "the boundary terms" in minimal area. If these terms are totally omitted, the remaining piece is regularization independent, but still differs by simple numerical factors like 4 from the double-loop integral which represents the BDS formula so that we stop short from the first non-trivial confirmation of the Alday-Maldacena duality. This confirms the earlier-found discrepancy for two parallel lines at n=infinity, but demonstrates that it actually affects only a finite number (out of infinitely many) of parameters in the functional dependence on the shape of the boundary, and the duality is only slightly violated, which allows one to call this violation an anomaly.