Euclidean Wilson loops and Minimal Area Surfaces in Minkowski AdS3

TL;DR

This paper explores minimal area surfaces in Minkowski AdS3 related to Euclidean Wilson loops, providing new analytical solutions and formulas for the area based on the Schwarzian derivative, advancing understanding in AdS/CFT correspondence.

Contribution

It generalizes previous Euclidean results to Minkowski AdS3, deriving a formula for the area and constructing an infinite family of analytical solutions using Riemann Theta functions.

Findings

Derived a formula for the area in terms of the Schwarzian derivative.

Constructed an infinite family of analytical solutions with Riemann Theta functions.

Validated the solutions by explicit calculations of surface area and shape.

Abstract

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5xS5 space. If the Wilson loop is Euclidean and confined to a plane (t,x) then the dual surface is Euclidean and lives in Minkowski AdS3. In this paper we study such minimal area surfaces generalizing previous results obtained in the Euclidean case. Since the surfaces we consider have the topology of a disk, the holonomy of the flat current vanishes which is equivalent to the condition that a certain boundary Schroedinger equation has all its solutions anti-periodic. If the potential for that Schroedinger equation is found then reconstructing the surface and finding the area become simpler. In particular we write a formula for the Area in terms of the Schwarzian derivative of the contour. Finally an infinite parameter family of analytical solutions using Riemann Theta functions is described. In this case, both the area and the shape of the surface are given analytically and used to check the previous results.