Grassmannian Geometries for Non-Planar On-Shell Diagrams

TL;DR

The paper extends the Grassmannian approach to non-planar on-shell diagrams, focusing on MHVs and the Grassmannian $G(2,n)$. It introduces the triplet formula to compute canonical functions $f_G$ via edge variables and a determinant construction, and decomposes these functions into Parke-Taylor factors through oriented regions. It proves that sphere moves, together with merge/expand moves, generate all identity moves for non-planar MHV diagrams, and shows that the associated geometries are often pseudo-positive, with internally planar irreducible diagrams yielding strongly connected, good-facet geometries. While planar diagrams sit inside the positive Grassmannian $G_+(2,n)$, non-planar cases generally require unions of oriented regions, which can be disconnected or possess null facets; some non-planar examples approach positivity, while others do not. The work lays groundwork for classifying non-planar Grassmannian geometries and suggests future exploration toward NMHV sectors and potential amplituhedron-like structures in the non-planar regime.

Abstract

On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

Figures (2)

• Figure 1: The 6-point sphere move. For convenience, in the bottom row we omit the external legs and label the white vertex they were attached to instead.
• Figure 2: A caricature of $R(12345) \cup R(12354)$.

Theorems & Definitions (20)

• Lemma 1: Non-vanishing condition
• proof
• Lemma 2: Factorization of the form
• proof
• Theorem 3
• proof
• Lemma 4: Pole for lone pair
• proof
• Lemma 5: Poles for irreducible hypertrees
• proof