Y-system for scattering amplitudes

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.

Notes on Euclidean Wilson loops and Riemann theta functions

“…Furthermore, the result gives, for each individual Wilson loop a one parameter family of deformation (given by the spectral parameter λ) such that the area remains the same. This is similar to what was observed in the case of the Wilson loops with light-like cusps [9]. Finally, for genus 3 we pointed out an interesting map between Wilson loops computable in this way and curves inside the Riemann surface.…”

Null-polygonal minimal surfaces in AdS4 from perturbed W minimal models

“…In particular, we derive the analytic expansion of the remainder function around the UV limit by using the underlying integrable models and the CPT. In this case, the corresponding TBA or Y-system is obtained by a projection from that for the minimal surfaces in AdS 5 [14]. The relevant CFT in the UV limit for the n-cusp surfaces is now the SU(n−4) 4 /U(1) n−5 generalized parafermion theory.…”

“…As studied in [13][14][15], such an area is governed by a set of non-linear integral equations of the TBA form or the associated T-/Y-systems. Those equations coincide with the TBA equations of the homogeneous sine-Gordon model.…”

“…At the boundary s = n − 4, we have also to impose the boundary condition related to the formal monodromy [14]. For n ∈ 4Z, the condition is simply given by…”

“…At strong coupling, the corresponding area of the minimal surfaces is evaluated with the help of integrability [12]. More concretely, one first solves a set of integral equations of the thermodynamic Bethe ansatz (TBA) form, or an associated Y-/T-system [13][14][15]. The cross-ratios are then expressed by its solution, i.e., the Y-or T-functions, and consequently the main part of the remainder function is given by these Y-/T-functions as well as the free energy associated with the TBA system.…”