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

TL;DR

This work surveys how integrable systems—via Pohlmeyer reduction, Hitchin systems, and TBA—provide a powerful route to compute the finite part of minimal-surface areas associated with scattering amplitudes/Wilson loops in AdS/CFT, bypassing direct nonlinear EOM solutions. Central to the approach are the Hitchin linear problems parametrized by $\zeta$, the Y-system for $A_{n-3}$, and the TBA equations whose free energy reproduces the nontrivial area contribution, with cross-ratios of the null polygon encoded in Y-functions. The review outlines how to reconstruct the classical string from linear problem data, define Fock–Goncharov coordinates on a WKB triangulation, and extract the minimal-area result from TBA data, while highlighting extensions to AdS$_5$, nonplanar amplitudes, and recent OPE/decomposition methods. Overall, the integrability-based program provides a concrete, nonperturbative handle on strong-coupling scattering in planar $N=4$ SYM, linking geometric boundary conditions to exact, solvable thermodynamic equations.

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.

Figures (4)

• Figure 1: 散射振幅与Wilson圈的对偶关系。左侧的D3膜延伸在空间$y^1,y^2,y^3$方向，并位于视界附近。在T-对偶后，我们得到了以类光多边形的Wilson圈为边界的世界面 Figure 1: Scattering amplitude/Wilson loop duality. The D3-brane on the left extends along the spatial directions $y^1, y^2, y^3$ and is located near the horizon. After T-duality, we obtain a worldsheet bounded by a light-like polygonal Wilson loop.
• Figure 2: $n=5$ 及$n=6$ 的WKB曲线以及对应的WKB三角分割。这里叉号代表$p(z)$的零点。简单起见，我们令所有的零点为实数。另外，我们对$\zeta$做了微小的转动，以避免连结零点和零点的WKB曲线。 Figure 2: The WKB curves and corresponding WKB triangulations for $n=5$ and $n=6$ . Here, the cross marks represent the zeros of $p(z)$. For simplicity, we assume all zeros to be real. Additionally, a slight rotation of $\zeta$ is applied to avoid WKB curves connecting zeros to zeros.
• Figure 3: 圈$\gamma_i$的构造。叉号代表黎曼面上的割点，波浪线是割线。 Figure 3: The construction of the cycle $\gamma_i$. The cross marks represent branch points on the Riemann surface, and the wavy lines denote branch cuts.
• Figure 4: 五边形OPE中的Wilson圈分解。 Figure 4: Wilson loop decomposition in pentagon OPE.