Minimal area surfaces in AdS n+1 and Wilson loops

Abstract

The AdS/CFT correspondence relates the expectation value of Wilson loops in \( \mathcal{N} \) = 4 SYM to the area of minimal surfaces in AdS₅. In this paper we consider minimal area surfaces in generic Euclidean AdS_n+1 using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS₃. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.