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.
Bourjaily, Jacob L.; Franco, Sebastian; Galloni, Danielle; and Wen, Congkao, "Stratifying on-shell cluster varieties: the geometry of non-planar on-shell diagrams" (2016). CUNY Academic Works.