Series Solution and Minimal Surfaces in AdS

TL;DR

This paper develops a series method to solve generalized Toda equations in AdS spaces, crucial for understanding minimal surfaces related to scattering amplitudes in Super Yang-Mills theory, especially for dimensions d>=4.

Contribution

The paper introduces a new series solution approach for generalized Toda equations in higher-dimensional AdS spaces, extending previous exact solutions known for AdS_3.

Findings

Series method accurately approximates solutions for generalized Toda equations.

Method agrees well with known exact solutions in AdS_3 case.

Provides insights into minimal surface areas relevant for scattering amplitudes.

Abstract

According to the Alday-Maldacena program the strong coupling limit of Super Yang-Mills scattering amplitudes is given by minimal area surfaces in AdS spacetime with a boundary consisting of a momentum space polygon. The string equations in AdS systematically reduce to coupled Toda type equations whose Euclidean classical solutions are then of direct relevance. While in the simplest case of AdS_3 exact solutions were known from earlier studies of the sinh-Gordon equation, there exist at present no similar exact forms for the generalized Toda equations related to AdS_d with d>=4. In this paper we develop a series method for the solution to those equations and evaluate their contribution to the finite piece of the worldsheet area. For the known sinh-Gordon case the method is seen to give results in excellent agreement with the exact answer.