Elliptical Wilson loops in ${\cal N}=4$ Super Yang-Mills

We investigate elliptical Wilson loops in ${\cal N}=4$ Super Yang--Mills theory at weak and strong coupling for small values of the eccentricity. We obtain analytical results for the vacuum expectation value of the Wilson loop in the form of a series in the eccentricity parameter. At weak coupling, we use perturbation theory in ${\cal N}=4$ Super Yang--Mills. At strong coupling, we use the AdS/CFT correspondence, which maps the Wilson loop to the minimal-area worldsheet of an open string in AdS space. We present a novel perturbative method to solve the Nambu--Goto equations allowing us to describe the minimal surface in terms of a coordinate parameterization in Euclidean AdS$_3$. Our results for the regularized area agree with those obtained by Dekel in [1] based on the Polyakov action.