Cyclicity of all anti-NMHV and hbox {N}^{2}{mathrm{MHV}} tree amplitudes in mathcal {N}=4 SYM

Abstract

This article proves the cyclicity of anti-NMHV and hbox {N}^{2}{mathrm{MHV}} tree amplitudes in planar mathcal {N}=4 SYM up to any number of external particles as an interesting application of positive Grassmannian geometry. In this proof the two-fold simplex-like structures of tree amplitudes introduced in 1609.08627 play a key role, as the cyclicity of amplitudes will induce similar simplex-like structures for the boundary generators of homological identities. For this purpose, we only need a part of all distinct boundary generators, and the relevant identities only involve BCFW-like cells. The manifest cyclic invariance in this geometric representation reflects one of the invariant characteristics of amplitudes, though they are obtained by the scheme-dependent BCFW recursion relation.