Cluster structures on spinor helicity and momentum twistor varieties

TL;DR

This paper explores the cluster algebra structures of spinor helicity and momentum twistor varieties, showing how they relate to Grassmannians and partial flag varieties, with implications for understanding scattering amplitudes.

Contribution

It demonstrates an embedding of partial flag variety coordinate rings into Grassmannian rings that respects cluster algebra structures, linking different varieties used in scattering amplitude theory.

Findings

Cluster algebra structures are preserved under the embedding.

Partial flag varieties can be embedded into Grassmannians respecting cluster seeds.

The relation between spinor helicity and momentum twistor varieties is explicitly characterized.

Abstract

We study the homogeneous coordinate rings of partial flag varieties and Grassmannians in their Pl\"ucker embeddings and exhibit an embedding of the former into the latter. Both rings are cluster algebras and the embedding respects the cluster algebra structures in the sense that there exists a seed for the Grassmannian that restricts to a seed for the partial flag variety (\textit{i.e.} it is obtained by freezing and deleting some cluster variables). The motivation for this project stems from the application of cluster algebras in scattering amplitudes: spinor helicity and momentum twistor varieties describe massless scattering without assuming dual conformal symmetry. Both may be obtained from Grassmanninas which model the dual conformal case. They are instances of partial flag varieties and their cluster structures reveal information for the scattering amplitudes. As an application of our main result we exhibit the relation between these cluster algebras.