Cyclicity of All Anti-NMHV and N$^2$MHV Tree Amplitudes in N=4 SYM

TL;DR

This paper proves the cyclicity of anti-NMHV and N$^2$MHV tree amplitudes in planar N=4 SYM using positive Grassmannian geometry, revealing geometric structures underlying amplitude invariance.

Contribution

It demonstrates the cyclicity of specific tree amplitudes in N=4 SYM through a geometric proof based on positive Grassmannian and simplex-like structures.

Findings

Cyclicity holds for all external particles in the studied amplitudes.

Simplex-like structures relate to boundary generators and homological identities.

Manifest cyclic invariance reflects fundamental amplitude characteristics.

Abstract

This article proves the cyclicity of anti-NMHV and N$^2$MHV tree amplitudes in planar 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.