Y-system for Scattering Amplitudes

TL;DR

This work develops a complete integrability-based framework to compute strong-coupling planar scattering amplitudes in N=4 SYM via minimal surfaces in AdS. By formulating flat connections with a spectral parameter and using Hitchin system techniques, the authors derive a Y-system whose thermodynamic Bethe Ansatz form yields the regularized area as the free energy, encoding all cross-ratio data of null polygon boundaries. The construction is first illustrated in AdS3 as a simpler SU(2) Hitchin reduction and then extended to AdS5, where a large Y/T lattice captures the full set of cross ratios, including connections to momentum twistors and zeta-symmetry. The results include explicit integral equations, area decompositions, and high-temperature limits corresponding to regular polygons, with reductions to AdS4/AdS3 and detailed analytic properties, providing a robust strong-coupling description of scattering amplitudes. This framework opens avenues for analytic continuation, excited-state generalizations, and potential quantum extensions of the AdS/CFT amplitudes at strong coupling.

Abstract

We compute N=4 Super Yang Mills planar amplitudes at strong coupling by considering minimal surfaces in AdS_5 space. The surfaces end on a null polygonal contour at the boundary of AdS. We show how to compute the area of the surfaces as a function of the conformal cross ratios characterizing the polygon at the boundary. We reduce the problem to a simple set of functional equations for the cross ratios as functions of the spectral parameter. These equations have the form of Thermodynamic Bethe Ansatz equations. The area is the free energy of the TBA system. We consider any number of gluons and in any kinematic configuration.

Figures (19)

• Figure 1: The polygon is specified at the $AdS$ boundary by the positions of the cusps ${\bf x}_i$. These positions are related to an ordered sequence of momenta ${\bf k}_i$ by ${\bf k}_i = {\bf x}_{i} - {\bf x}_{i-1}$. The two dimensional minimal surface streches in the $AdS$ bulk and ends on the polygonal contour at the boundary.
• Figure 2: Spacetime positions of the cusps for a polygon that is embedded in $R^{1,1}$, which is the boundary of $AdS_3$. The positions of the cusps are given by a set of $n/2$ values $x^+_i$ and a set of $n/2$ values $x^-_i$.
• Figure 3: In this figure we have summarized the structure of the $T$'s and the $Y$'s in a gauge where we simplified the $T$'s that can be simplified. The small solid black dots represent non-zero $T$-functions. They are equal to one unless $a=1$ and $s=1,\dots,n/2-2$. At the rightmost point in this line we have $T_{1,n/2-2}=B$ where $B$ is a function which cannot be set to one only in the case that $n/2$ is even. In fact, in our case it is $B =- e^{ m/\zeta + { \bar{m} }\zeta}$. This is the function that governs the monodormy $s_{n/2} = - B(\zeta e^{ i \pi (n/2 +1)/2} ) s_0$. The $Y$-functions are finite in the points indicated by fat gray shaded balls. At all other points they are either zero or infinity.
• Figure 4: Sketch of the pattern of WKB lines for various phases of $\zeta$. The crosses denote the various zeros of $p(z)$. The numbers indicate the various Stokes sectors. The black thin lines end on the zeros and separate different classes of WKB lines. The thick colored lines are the WKB lines that we use to evaluate cross ratios. Here we have indicated only the ones used to evaluate $Y_{2}$ and $Y_3$. Finally, note that by setting the phase of $\zeta$ to $e^{ i \pi/4}$ we have WKB lines that enable us to evaluate all the $Y_s$.
• Figure 5: Cycles along which we need to integrate $\sqrt{p} dz$ in order to determine the asymptotic form of $Y_s$. By being careful about the sheet selected by the various small solutions one can determine the cycle orientations shown here.