The Grassmannian Origin Of Dual Superconformal Invariance

A dual formulation of the S Matrix for N = 4 SYM has recently been presented, where all leading singularities of n-particle Nk−2MHV amplitudes are given as an integral over the Grassmannian G(k, n), with cyclic symmetry, parity and superconformal invariance manifest. In this short note we show that the dual superconformal invariance of this object is also manifest. The geometry naturally suggests a partial integration and simple change of variable to an integral over G(k − 2, n). This change of variable precisely corresponds to the mapping between usual momentum variables and the “momentum twistors” introduced by Hodges, and yields an elementary derivation of the momentum-twistor space formula very recently presented by Mason and Skinner, which is manifestly dual superconformal invariant. Thus the G(k, n) Grassmannian formulation allows a direct understanding of all the important symmetries of N = 4 SYM scattering amplitudes. Recently a simple formula has been conjectured for all leading singularities of N = 4 SYM amplitudes [1]. Working in twistor space, the leading singularities of single trace, color stripped, n-particle NMHV amplitudes are associated with the object Ln;k = 1 vol(GL(k)) ∫ dCαa (C1C2 · · ·Ck) · · · (CnC1 · · ·Ck−1) k