On n-point Amplitudes in N=4 SYM

TL;DR

This paper explores the computation of n-point amplitudes in N=4 SYM at strong coupling, linking integrable sigma-model solutions to geometric boundary integrals, and constructs a family of solutions for the four-point case.

Contribution

It constructs a multi-parameter family of solutions for the four-point amplitude and analyzes their regularized areas, highlighting the geometric interpretation of the amplitude.

Findings

Multiple solutions correspond to the same kinematic invariants.

The minimal regularized area matches the Alday-Maldacena solution.

A geometric duality between minimal area and boundary contour integrals is confirmed.

Abstract

The computation of n-point planar amplitudes in N=4 SYM at strong coupling is known to be reduced to the search for solutions of the integrable 2d SO(4,2) sigma-model with growing asymptotics on the world-sheet and to the study of their Whitham deformations induced by an epsilon-regularization, which breaks both integrability and SO(4,2) symmetry. A multi-parameter (moduli) family of such solutions is constructed for n=4. They all correspond to the same s and t and some are related by SO(4,2) transformations. Nevertheless, they lead to different regularized areas, whose minimum is the Alday-Maldacena solution. A brief review of results on n-point amplitudes is also provided, with special emphasis on the underlying equivalence of the above regularized minimal area in AdS and a double contour integral along the same boundary, two purely geometric quantities.