Cluster structures on spinor helicity and momentum twistor varieties

TL;DR

This work establishes a unifying cluster-algebra framework linking partial flag varieties and Grassmannians via Plücker embeddings. It proves that the homogeneous coordinate rings of partial flags embed into Grassmannian coordinate rings in a way compatible with cluster structures: the initial seed of a flag is a restricted seed of a Grassmannian seed, obtainable by explicit mutation sequences. The authors develop a tableaux-based dictionary for cluster variables, connecting dual canonical bases to semistandard tableaux, and provide mutation recipes that realize these correspondences. As applications, momentum twistor varieties MT$_n$ and spinor helicity varieties SH$_n$ inherit cluster structures from Grassmannians, enabling concrete mutation sequences and seed descriptions that are relevant for scattering-amplitude program and dual conformal symmetry analyses.

Abstract

We study the homogeneous coordinate rings of partial flag varieties and Grassmannians in their Plücker 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.

Figures (15)

• Figure 1: The pseudoline arrangement $\mathcal{P}_{2,5;7}$ and its quiver $Q_{2,5;7}$ together with the index sets $I_F$ of the initial minors and the additional frozen vertices corresponding to $d_1=2,d_2=5$ labelled by their weights (as is §\ref{['sec:U']}), $\omega_2$ respectively $\omega_5$.
• Figure 2: The shading of the regions indicates the weight of the associated minor. Notice that the vertices $\bigstar$ and $\blacklozenge$ are non-balanced while all others are balanced.
• Figure 3: The initial seeds for the partial flag varieties $\mathop{\mathrm{\mathcal{F}\space\ell}}\nolimits_{2,4;5}$ (on the left) and $\mathop{\mathrm{\mathcal{F}\space\ell}}\nolimits_{2,4;6}$ (on the right).
• Figure 4: The initial seed for the partial flag variety $\mathop{\mathrm{\mathcal{F}\space\ell}}\nolimits_{2,5;7}$.
• Figure 5: The initial seed for ${\mathbb{C}}[\mathop{\mathrm{Gr}}\nolimits_{k;n}]$ we used.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Example 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4