Grassmannian Geometry And Simplex-like Structures Of Scattering Amplitudes

This article revisits and elaborates the significant role of positive geometry of momentum twistor Grassmannian for planar N= 4 SYM scattering amplitudes.First we establish the fundamentals of positive Grassmannian geometry for tree amplitudes, including the ubiquitous Pliicker coordinates and the representation of reduced Grassmannian geometry. Then we formulate this subject, without making reference to on-shell diagrams and decorated permutations, around these four major facets: 1. Deriving the tree and 1-loop BCFW recursion relations solely from positivity, after introducing the simple building blocks called positive com-ponents for a positive matrix. 2. Applying Grassmannian geometry and Pliicker coordinates to determine the signs of N2MHV homological identities, which inter-connect various Yangian invariants. It reveals that most of the signs are in fact the secret incarnation of the simple 6-term NMHV identity. 3. Deriving the stacking positivity relation, which is powerful for parameterizing matrix representatives in terms of positive variables in the dlog form. It will be used with the reduced Grassmannian geometry representation to produce the positive matrix of a given geometric configuration, which is an independent approach besides the combina-toric way involving a sequence of BCFW bridges. 4. Introducing an elegant and highly refined formalism of BCFW recursion relation for tree amplitudes, which reveals its two-fold simplex-like structures. First, the BCFW contour in terms of (reduced) Grassmannian geometry representatives is delicately dissected into a triangle-shape sum, as only a small fraction of the sum needs to be explicitly identified. Second, this fraction can be further dissected, according to different growing modes with corresponding growing parameters. The growing modes pos-sess the shapes of solid simplices of various dimensions, with which infinite number of BCFW cells can be entirely captured by the characteristic objects called fully-spanning cells. We find that for a given k, beyond n = 4k + 1 there is no more new fully-spanning cell, which signifies the essential termination of the recursivegrowth of BCFW cells. As n increases beyond the termination point, the BCFW contour simply replicates itself according to the simplex-like patterns, which en-ables us to master all BCFW cells once for all without actually identifying most of them.