# Asymptotic AdS String Solutions for Null Polygonal Wilson Loops in R^{1,2}

## TL;DR

This paper extends asymptotic AdS_3 string solutions to AdS_4 coordinates, demonstrating closed null polygonal Wilson loops and constructing specific solutions for tetragonal loops, revealing connections with conformal transformations.

## Contribution

It expresses asymptotic AdS_3 string solutions in AdS_4 coordinates and constructs new solutions for tetragonal Wilson loops with parameter freedom linked to conformal symmetry.

## Key findings

- Hexagonal and octagonal Wilson loops form closed null contours.
- Constructed a two-parameter string solution for tetragonal Wilson loops.
- Parameter freedoms relate to conformal SO(2,4) transformations.

## Abstract

For the asymptotic string solution in AdS_3 which is represented by the AdS_3 Poincare coordinates and yields the planar multi-gluon scattering amplitude at strong coupling in arXiv:0904.0663, we express it by the AdS_4 Poincare coordinates and demonstrate that the hexagonal and octagonal Wilson loops surrounding the string surfaces take closed contours consisting of null vectors in R^{1,2} owing to the relations of Stokes matrices. For the tetragonal Wilson loop we construct a string solution characterized by two parameters by solving the auxiliary linear problems and demanding a reality condition, and analyze the asymptotic behavior of the solution in R^{1,2}. The freedoms of two parameters are related with some conformal SO(2,4) transformations.