Non-planar BCFW Grassmannian Geometries

TL;DR

This paper explores non-planar Grassmannian geometries related to non-adjacent BCFW recursion in ${\

Contribution

It introduces a generalization of positive Grassmannian cells to non-planar geometries for non-adjacent BCFW shifts, expanding the geometric understanding of scattering amplitudes.

Findings

Non-adjacent BCFW shifts lead to non-planar Grassmannian geometries.

These geometries generalize the positive Grassmannian to a larger class of objects.

A new method for calculating on-shell functions using Grassmannian configurations is proposed.

Abstract

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the $n$-point N$^k$MHV tree-level amplitude in ${\cal N}=4$ SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical dlog forms on the cells in the positive Grassmannian $G_+(k,n)$. Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian $G_+(k,n)$ to a larger class of objects which live in $G(k,n)$. We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.