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

We investigate elliptical Wilson loops in $\mathcal{N}=4$ 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 $\mathcal{N}=4$ 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.