HET seminar: Matteo Parisi
Speaker: Matteo Parisi
Title: Amplituhedra: Scattering Amplitudes from Geometry
Abstract: The Amplituhedra A(n,k,m) are generalisations of polytopes introduced as a geometric construction encoding scattering amplitudes in N=4 supersymmetric Yang-Mills theory (SYM). These are extracted from a differential form, the canonical form of the Amplituhedron, which emerges from a purely geometric definition. Following my recent works, I will explain how the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry, computes the canonical form for whole families of objects, namely for Amplituhedra of type A(n,1,m), which are cyclic polytopes and for their conjugates A(n,n-m-1,m) for even m, which are not polytopes. This method connects to the rich combinatorial structure of triangulations of Amplituhedra, captured by what we refer to as ‘Secondary Geometry’. For polygons, this is the `Associahedron', explored by Stasheff in the sixties; for polytopes, it is the `secondary polytope' constructed by the Gelfand's school in the nineties. Whereas, for Amplituhedra, we are the first to initiate the studies of what we called the ‘Secondary Amplituhedra’. The latter encodes all representations of scattering amplitudes, many not obtainable with any physical method, together with their algebraic relations produced by global residue theorems. Finally, I will briefly illustrate some of the recent geometric directions in my work on the Amplituhedron in momentum space and new exciting developments connecting the (secondary geometry of) m=2 amplituhedron with the positive tropical Grassmannian. This object has been appearing in dozens of papers in the physics community in the last year, both in bootstrapping loop amplitudes in planar N=4 SYM and in computing (a generalisation of) biadjoint scalar amplitudes.