Scattering Amplitudes and the Positive Grassmannian

摘要: We establish a direct connection between scattering amplitudes in planar four-dimensional theories and remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams objects of fundamental importance amplitudes. show that all-loop integrand N=4 SYM naturally represented this way. On-shell theory are intimately tied variety objects, ranging from new graphical representation permutations beautiful stratification Grassmannian G(k,n) which generalizes notion simplex projective space. All physically important operations involving map canonical permutations; particular, BCFW deformations correspond adjacent transpositions. Each cell endowed with coordinates an invariant measure determines function associated diagram. This understanding allows us classify compute all diagrams, give geometric for non-trivial relations among them. Yangian invariance transparently by diffeomorphisms preserve structure. Scattering (1+1)-dimensional integrable systems ABJM (2+1) dimensions can both be understood special cases these ideas. less (or no) supersymmetry exactly same structures Grassmannian, but deformed factor encoding ultraviolet singularities. processes also gives amplitudes, presenting integrands novel dLog form directly reflects underlying