Local Spacetime Physics from the Grassmannian

TL;DR

This paper demonstrates how a contour deformation in the Grassmannian integral framework connects different amplitude expansions in N=4 SYM, revealing the emergence of local spacetime physics from abstract geometric structures.

Contribution

It shows that a simple contour deformation relates Grassmannian residues to the CSW and Risager expansions, linking geometric and spacetime descriptions of scattering amplitudes.

Findings

Contour deformation converts Grassmannian residues to CSW expansion.

Risager degrees of freedom are determined by Grassmannian gauge choices.

The approach reveals local spacetime physics from Grassmannian geometry.

Abstract

A duality has recently been conjectured between all leading singularities of n-particle N^(k-2)MHV scattering amplitudes in N=4 SYM and the residues of a contour integral with a natural measure over the Grassmannian G(k,n). In this note we show that a simple contour deformation converts the sum of Grassmannian residues associated with the BCFW expansion of NMHV tree amplitudes to the CSW expansion of the same amplitude. We propose that for general k the same deformation yields the (k-2) parameter Risager expansion. We establish this equivalence for all MHV-bar amplitudes and show that the Risager degrees of freedom are non-trivially determined by the GL(k-2) "gauge" degrees of freedom in the Grassmannian. The Risager expansion is known to recursively construct the CSW expansion for all tree amplitudes, and given that the CSW expansion follows directly from the (super) Yang-Mills Lagrangian in light-cone gauge, this contour deformation allows us to directly see the emergence of local space-time physics from the Grassmannian.