Scattering Amplitudes and the Positive Grassmannian

TL;DR

This work uncovers a direct, unifying framework linking planar four-dimensional scattering amplitudes to the positive Grassmannian via on-shell diagrams. By representing amplitudes as canonical differential forms on G(k,n) and exploiting positroid stratification, it reveals how BCFW-type recursions, Yangian symmetry, and dual conformal invariance emerge geometrically. The construction gives a d log representation of all-loop integrands, a natural positivity structure, and a rich interplay with cluster algebras and momentum twistors, extending to ABJM and integrable systems. The approach yields a powerful, coordinate-friendly dictionary between physical singularities, combinatorics of decorated permutations, and high-dimensional geometry, with broad implications for understanding locality, unitarity, and transcendentality in quantum field theory.

Abstract

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes, presenting all integrands in a novel dLog form which directly reflects the underlying positive structure.