Sign Flip Triangulations of the Amplituhedron
Abstract
We present new triangulations of the m=4 amplituhedron relevant for scattering amplitudes in planar N=4 super-Yang-Mills, obtained directly from the combinatorial definition of the geometry. Using the "sign flip" characterization of the amplituhedron, we reproduce the canonical forms for the all-multiplicity next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (N2MHV) tree-level as well as the NMHV one-loop cases, without using any input from traditional amplitudes methods. Our results provide strong evidence for the equivalence of the original definition of the amplituhedron Arkani-Hamed:2013jha and the topological one Arkani-Hamed:2017vfh, and suggest a new path forward for computing higher loop amplitudes geometrically. In particular, we realize the NMHV one-loop amplituhedron as the intersection of two amplituhedra of lower dimensionality, which is reflected in the novel structure of the corresponding canonical form.