A Grassmannian Etude in NMHV Minors

TL;DR

Problem: establish equivalence between the linked-connected twistor-string prescription and the ACCK Grassmannian integrand for all tree-level NMHV amplitudes in $\mathcal{N}=4$ SYM. Approach: apply the multidimensional Global Residue Theorem to map the connected contour to the ACCK BCFW contour, with explicit checks at $n=6$ and $n=7$ and a general all-$n$ proof. Key contributions include showing a simple sextic deformation leaves the amplitude invariant and in the zero-deformation limit yields the ACCK integrand, plus a four-channel residue decomposition corresponding to different sextic structures. Significance: strengthens the geometric link between link variables, Grassmannians, and contour integrals for NMHV amplitudes and clarifies the role of BCFW-type residues in this framework.

Abstract

Arkani-Hamed, Cachazo, Cheung and Kaplan have proposed a Grassmannian formulation for the S-matrix of N=4 Yang-Mills as an integral over link variables. In parallel work, the connected prescription for computing tree amplitudes in Witten's twistor string theory has also been written in terms of link variables. In this paper we extend the six- and seven-point results of arXiv:0909.0229 and arXiv:0909.0499 by providing a simple analytic proof of the equivalence between the two formulas for all tree-level NMHV superamplitudes. Also we note that a simple deformation of the connected prescription integrand gives directly the ACCK Grassmannian integrand in the limit when the deformation parameters equal zero.