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Minimal area surfaces dual to Wilson loops and the Mathieu equation

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 $λ$-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 $AdS_3$.