Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams

TL;DR

This paper systematically classifies non-planar on-shell varieties in the Grassmannian, revealing new geometric features and extending the understanding of on-shell diagrams beyond the planar limit.

Contribution

It provides a complete classification of on-shell varieties in G(3,6), highlighting differences from planar cases and proposing a framework for studying non-planar on-shell geometries.

Findings

24 top-dimensional varieties identified in G(3,6)

10 co-dimension one varieties found in G(3,6)

Distinct features of non-planar varieties compared to positroids

Abstract

The correspondence between on-shell diagrams in maximally supersymmetric Yang-Mills theory and cluster varieties in the Grassmannian remains largely unexplored beyond the planar limit. In this article, we describe a systematic program to survey such 'on-shell varieties', and use this to provide a complete classification in the case of $G(3,6)$. In particular, we find exactly 24 top-dimensional varieties and 10 co-dimension one varieties in $G(3,6)$---up to parity and relabeling of the external legs. We use this case to illustrate some of the novelties found for non-planar varieties relative to the case of positroids, and describe some of the features that we expect to hold more generally.