Local Spacetime Physics from the Grassmannian
Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Jaroslav Trnka
TL;DR
This work shows that a canonical contour deformation of the Grassmannian representation of N=4 SYM amplitudes, achieved by relaxing a delta-function, converts the BCFW-like sum over residues into the CSW expansion for NMHV amplitudes and, more generally, into the Risager expansion for higher N^k−2MHV amplitudes. The NMHV CSW equivalence is demonstrated explicitly, with nonlocal residues canceling and CSW terms emerging as the localized, spacetime-relevant contributions. Moreover, the Risager deformation parameters (the (k−2) degrees of freedom) naturally arise from the GL(k−2) gauge freedom in the momentum-twistor Grassmannian, revealing a direct link between gauge redundancy and spacetime locality. Altogether, the paper unifies Grassmannian residues with light-cone gauge Lagrangian physics, providing a transparent mechanism by which local spacetime structure emerges from a manifestly symmetric integral representation and suggesting paths to extend to loop-level leading singularities and twistor-string connections.
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.