Minimal area surfaces in AdS_{n+1} and Wilson loops

TL;DR

This paper extends the study of minimal area surfaces in Euclidean AdS spaces to higher dimensions using Pohlmeyer reduction, linking boundary curve invariants to minimal surface areas and exploring eformation symmetries.

Contribution

It generalizes the Pohlmeyer reduction approach to Euclidean AdS_{n+1} and introduces a method to parameterize boundary curves via conformal invariants, enabling analysis of eformed contours.

Findings

Expressed minimal surface area as boundary integrals involving conformal invariants.

Established a eformation symmetry for boundary contours.

Linked periodicity of eformed contours to conserved charges from integrability.

Abstract

The AdS/CFT correspondence relates the expectation value of Wilson loops in N=4 SYM to the area of minimal surfaces in AdS_5 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 AdS_3. 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 \lambda-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the \lambda-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all \lambda 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.