Thermodynamic Bethe ansatz and scattering amplitude

In this review, 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 (AdS) space. 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 work 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 theY-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.