Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space

TL;DR

This paper studies minimal surfaces in AdS space ending on null polygonal boundaries, linking the problem to integrable equations and applying results to compute areas related to gluon scattering amplitudes.

Contribution

It connects the minimal surface problem in AdS to Hitchin equations and integrable systems, providing explicit solutions for polygonal boundaries and their areas.

Findings

Explicit area calculations for an eight-sided polygon.

Solutions for regular polygonal boundaries.

Connection to gluon scattering amplitudes at strong coupling.

Abstract

We consider minimal surfaces in three dimensional anti-de-Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. We describe in detail the mathematical problem that needs to be solved. This problem is mathematically the same as the one studied by Gaiotto, Moore and Neitzke in the context of the moduli space of certain supersymmetric theories. Using their results we can find the explicit answer for the area of a surface that ends on an eight-sided polygon. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling. In addition, we give fairly explicit solutions for regular polygons.