Minimal area surfaces dual to Wilson loops and the Mathieu equation

TL;DR

This paper introduces a new class of analytically solvable minimal area surfaces in AdS space, linked to Wilson loops, using the Mathieu equation, and explores their properties and limits within the AdS/CFT framework.

Contribution

It applies the Mathieu equation to construct and analyze minimal surfaces dual to Wilson loops, revealing new solutions with umbilical points and invariance under deformations.

Findings

Analytic expressions for minimal surface areas using Mathieu eigenvalues.

Identification of Wilson loops with umbilical points and their deformation properties.

Connections to circular and light-like polygon Wilson loops in various limits.

Abstract

The AdS/CFT correspondence relates Wilson loops in N=4 SYM to minimal area surfaces in $AdS_5\times S^5$ space. Recently, a new approach to study minimal area surfaces in $AdS_3 \subset AdS_5$ was discussed based on a Schroedinger equation with a periodic potential determined by the Schwarzian derivative of the shape of the Wilson loop. Here we use the Mathieu equation, a standard example of a periodic potential, to obtain a class of Wilson loops such that the area of the dual minimal area surface can be computed analytically in terms of eigenvalues of such equation. As opposed to previous examples, these minimal surfaces have an umbilical point (where the principal curvatures are equal) and are invariant under $\lambda$-deformations. In various limits they reduce to the single and multiple wound circular Wilson loop and to the regular light-like polygons studied by Alday and Maldacena. In this last limit, the periodic potential becomes a series of deep wells each related to a light-like segment. Small corrections are described by a tight--binding approximation. In the circular limit they are well approximated by an expansion developed by A.Dekel. In the particular case of no umbilical points they reduce to a previous solution proposed by J. Toledo. The construction works both in Euclidean and Minkowski signature of $AdS_3$.