Scattering Amplitudes and Toric Geometry

TL;DR

This paper explores a novel toric geometric framework for understanding scattering amplitudes in planar N=4 SYM, linking on-shell diagrams, Grassmannians, and toric varieties to reveal new structural insights.

Contribution

It introduces a toric geometric interpretation of scattering amplitudes, connecting on-shell diagrams and Grassmannians within a new algebraic variety framework.

Findings

Toric varieties associated with on-shell diagrams are described in detail.

Singularities of amplitudes are encoded in subspaces of toric varieties.

The action of cluster transformations on toric varieties is analyzed.

Abstract

In this paper we provide a first attempt towards a toric geometric interpretation of scattering amplitudes. In recent investigations it has indeed been proposed that the all-loop integrand of planar N=4 SYM can be represented in terms of well defined finite objects called on-shell diagrams drawn on disks. Furthermore it has been shown that the physical information of on-shell diagrams is encoded in the geometry of auxiliary algebraic varieties called the totally non negative Grassmannians. In this new formulation the infinite dimensional symmetry of the theory is manifest and many results, that are quite tricky to obtain in terms of the standard Lagrangian formulation of the theory, are instead manifest. In this paper, elaborating on previous results, we provide another picture of the scattering amplitudes in terms of toric geometry. In particular we describe in detail the toric varieties associated to an on-shell diagram, how the singularities of the amplitudes are encoded in some subspaces of the toric variety, and how this picture maps onto the Grassmannian description. Eventually we discuss the action of cluster transformations on the toric varieties. The hope is to provide an alternative description of the scattering amplitudes that could contribute in the developing of this very interesting field of research.