# $\Delta$-Algebra and Scattering Amplitudes

## TL;DR

This paper introduces the $ ext{Delta}$-algebra, a mathematical framework that unifies polytope volume calculations and canonical form constructions in scattering amplitudes, leading to new moves and relations in on-shell diagrams.

## Contribution

It develops the $ ext{Delta}$-algebra for scattering amplitudes, generalizes the square move to sphere moves, and derives new higher-order relations, advancing the mathematical understanding of amplitude structures.

## Key findings

- Derived the $ ext{Delta}$-algebra and its applications.
- Generalized the square move to sphere moves for non-planar diagrams.
- Proved new formulas for higher-order amplitude relations.

## Abstract

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.