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

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.

💡 Research Summary

This review article surveys a powerful integrability‑based method for computing the area of minimal surfaces that appear in the scattering amplitude/Wilson loop duality of planar 𝒩=4 super‑Yang‑Mills theory. In the dual picture the strong‑coupling gluon scattering amplitude is given by the exponential of the regularised area of a string world‑sheet in AdS₅ whose boundary is a light‑like polygonal Wilson loop on the AdS boundary. Directly solving the nonlinear string equations of motion together with the Virasoro constraints is prohibitively difficult for polygons with more than four edges because of the intricate boundary conditions.

The authors bypass this difficulty by first performing a Pohlmeyer reduction of the classical string equations in AdS₃ (the essential features already appear there). The reduction rewrites the system as a modified sinh‑Gordon equation for a scalar field α(z, \bar z) coupled to a holomorphic function p(z). This pair of fields is equivalent to a Hitchin system – a flat SL(2,ℂ) connection depending on a complex spectral parameter ζ. The flatness condition reproduces the modified sinh‑Gordon equation, while the linear problem associated with the connection provides two independent solutions (the “large” and “small” solutions).

Boundary data for the minimal surface are encoded in the Stokes phenomena of the small solutions. By prescribing the asymptotic behaviour of p(z) (a polynomial of degree n‑2) and the growth of α(z) at infinity, one reproduces exactly the geometry of an n‑sided light‑like polygon. The small solutions generate SL(2)‑invariant inner products s_i∧s_j that define cross‑ratios χ_{ijkl}, which are the conformally invariant data of the Wilson loop.

The next step is to construct Fock‑Goncharov coordinates X_E from the inner products of four small solutions attached to each edge E of a WKB triangulation of the complex z‑plane. The WKB triangulation is obtained by drawing the Stokes lines (or “WKB curves”) associated with the differential λ² = p(z)dz²; each triangle corresponds to a sector where a particular small solution dominates. The X_E variables satisfy a set of functional relations known as the T‑system. By a standard change of variables one arrives at a Y‑system Y_{a,s}(θ) that obeys a set of coupled nonlinear integral equations – the Thermodynamic Bethe Ansatz (TBA) equations.

The TBA equations have the generic form

  log Y_{a,s}(θ) = – m_{a,s} cosh θ + ∑{b,t} K{a,s}^{b,t} * log(1+Y_{b,t}(θ)) ,

where * denotes convolution in rapidity θ, m_{a,s} are mass parameters fixed by the polygon data, and K are known kernels. The free energy associated with this TBA system is

  F = – ∑{a,s} ∫ (dθ/2π) m{a,s} cosh θ log(1+Y_{a,s}(θ)) .

A crucial result is that this free energy equals the non‑trivial (regularised) part of the minimal surface area, i.e. A_non‑trivial = F. Consequently, the problem of computing the amplitude reduces to solving the TBA integral equations, which can be tackled numerically or analytically in certain limits (e.g. large‑mass or near‑collinear regimes).

The review demonstrates the method explicitly for the four‑point case (reproducing the Alday‑Maldacena result) and discusses how the six‑point and higher‑point amplitudes deviate from the BDS ansatz. The deviation is precisely captured by the TBA free energy, confirming the integrability picture.

In the final section the authors outline several promising research directions: extending the boundary data to more general complex curves with irregular singularities, incorporating quantum corrections via the ODE/IM correspondence, exploring analogous Y‑systems in the full AdS₅/CFT₄ setting, and studying wall‑crossing phenomena that modify the Stokes data. These extensions aim to deepen our understanding of non‑perturbative scattering amplitudes, reveal new structures in the gauge/string correspondence, and potentially provide exact results for a broader class of observables in strongly coupled gauge theories.