[Paper Review] Scattering Amplitudes and the Positive Grassmannian

Bourjaily, Jacob|arXiv (Cornell University)|Jan 1, 2014

Black Holes and Theoretical PhysicsPhysics and Astronomy170 references271 citations

TL;DR

This paper establishes a direct geometric correspondence between scattering amplitudes in planar four-dimensional quantum field theories and the positive Grassmannian G(k,n). By treating on-shell diagrams as fundamental objects, it shows that all-loop integrands in N=4 SYM arise from canonical measures on cells of the positive Grassmannian, with BCFW shifts corresponding to adjacent transpositions on permutations and Yangian invariance encoded as positive structure-preserving diffeomorphisms. The key contribution is a unified dLog form for all-loop integrands that transparently reflects the underlying positive geometry.

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.

Motivation & Objective

• To establish a direct correspondence between scattering amplitudes in planar four-dimensional theories and the positive Grassmannian G(k,n).
• To show that on-shell diagrams in N=4 SYM naturally correspond to cells in the positive Grassmannian with canonical positive coordinates and invariant measures.
• To demonstrate that physical operations like BCFW deformations map to canonical operations on permutations, such as adjacent transpositions.
• To provide a geometric explanation for non-trivial relations among on-shell diagrams via the stratification of G(k,n).
• To extend the framework to (1+1)-dimensional integrable models and ABJM theory as special cases, and to less supersymmetric theories via deformed measures.

Proposed method

• Represent on-shell diagrams as combinatorial objects tied to permutations and their adjacent transpositions.
• Map each on-shell diagram to a cell in the positive Grassmannian G(k,n), assigning it positive coordinates and a canonical measure.
• Use the positive structure of G(k,n) to define an invariant measure that generates the on-shell function for each diagram.
• Apply diffeomorphisms preserving the positive structure to geometrically realize Yangian invariance of scattering amplitudes.
• Derive a dLog form for the all-loop integrand by expressing it in terms of canonical coordinates on the positive Grassmannian.
• Generalize the framework to non-supersymmetric and less supersymmetric theories by deforming the measure to encode UV singularities.

Experimental results

Research questions

• RQ1 How can on-shell diagrams in planar N=4 SYM be systematically classified using the positive Grassmannian?
• RQ2 What is the geometric origin of non-trivial relations among on-shell diagrams in scattering amplitudes?
• RQ3 How do BCFW deformations manifest within the positive Grassmannian framework?
• RQ4 In what way does Yangian invariance emerge from diffeomorphisms on G(k,n) that preserve the positive structure?
• RQ5 How can the dLog form of the all-loop integrand be derived directly from the positive Grassmannian geometry?

Key findings

• All-loop integrands in N=4 SYM are naturally expressed in a dLog form that directly reflects the positive geometry of the Grassmannian.
• On-shell diagrams correspond bijectively to cells in the positive Grassmannian G(k,n), each equipped with positive coordinates and an invariant measure.
• BCFW deformations are geometrically realized as adjacent transpositions on permutations associated with the diagrams.
• Yangian invariance is transparently encoded as diffeomorphisms of G(k,n) that preserve the positive structure.
• Scattering amplitudes in (1+1)-dimensional integrable systems and ABJM theory arise as special cases of the same Grassmannian framework.
• In non-supersymmetric or less supersymmetric theories, the same Grassmannian structure applies, but the measure is deformed to encode ultraviolet singularities.