A short review on TBA equation and scattering amplitude/Wilson loop duality

TL;DR

This review discusses the use of integrable systems, particularly the Hitchin system and TBA equations, to compute minimal surface areas in the scattering amplitude/Wilson loop duality within AdS/CFT, avoiding direct solutions of complex equations.

Contribution

It introduces an alternative integrable systems approach, including the Hitchin system and Y/TBA equations, to calculate minimal surfaces in the duality, offering new computational methods.

Findings

Boundary conditions for the Hitchin system are established.

Y-system and TBA equations are derived from the Hitchin system.

The free energy of TBA equations relates to the minimal surface area.

Abstract

In this review (written in Chinese), we introduce the computation of the minimal surface area in the scattering amplitude/Wilson loop duality, where the minimal surface ends on a light-like polygonal Wilson loop at the boundary of anti-de Sitter space (AdS). Due to its nonlinearity and the complexity of the boundary conditions, directly solving the equations of motion to compute the area is highly challenging. This paper reviews an alternative approach that bypasses the direct solution of the equations of motion and instead uses integrable systems to compute the area. We will provide boundary conditions for the Hitchin system, which is equivalent to the equations of motion, to describe the light-like polygonal boundary of the minimal surface. Starting from the solution of the Hitchin system, we will further derive the Y-system and the Thermodynamic Bethe Ansatz (TBA) equations, whose free energy provides the nontrivial part of the minimal surface area. Finally, we will discuss recent developments in this field and provide an outlook for future research.