Elliptical Wilson loops in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> Super Yang-Mills theory

We investigate elliptical Wilson loops in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>4</a:mn> </a:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi mathvariant="script">N</d:mi> <d:mo>=</d:mo> <d:mn>4</d:mn> </d:math> Super Yang–Mills. At strong coupling, we use the AdS/CFT correspondence, which maps the Wilson loop to the minimal-area world sheet of an open string in anti–de Sitter (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 parametrization in Euclidean <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>AdS</g:mi> </g:mrow> <g:mrow> <g:mn>3</g:mn> </g:mrow> </g:msub> </g:mrow> </g:math> . Our results for the regularized area agree with those obtained by Dekel [.] based on the Polyakov action.