Towards the Amplituhedron Volume

TL;DR

This paper explores the geometric structure of scattering amplitudes in planar N=4 super Yang-Mills theory by connecting the amplituhedron volume to differential equations, offering a new triangulation-free approach.

Contribution

It introduces a novel differential equation framework to compute amplituhedron volumes for NMHV amplitudes, simplifying the geometric understanding.

Findings

Amplituhedron volume for NMHV amplitudes is determined by specific differential equations.

The new formulation provides a straightforward geometric description without triangulations.

Discussion of potential extensions to N^kMHV amplitudes volumes.

Abstract

It has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.