Matteo Parisi

Amplituhedra: Jeffrey-Kirwan Residue and Secondary Geometry

The Amplituhedra A(n,k,m) are Grassmannian generalisations of polytopes. They have been recently introduced by physicists as a geometric construction encoding interactions of fundamental particles in Quantum Field Theory. In particular, scattering amplitudes in N=4 supersymmetric Yang-Mills theory are extracted from a differential form, the canonical form of the Amplituhedron, which emerges from a purely geometric definition.

I will introduce the amplituhedra, describing them in various special cases, and explaining their relation with scattering amplitudes.

As main result of our work, I will then show how the Jeffrey-Kirwan Residue, a powerful concept in Symplectic and Algebraic Geometry, computes the canonical form for whole families of objects: the Amplituhedra of type A(n,1,m), which are cyclic polytopes, and their conjugates A(n,n-m-1,m) for even m, which are not polytopes.

Finally, I will explain how this method connects to the rich combinatorial structure of triangulations of Amplituhedra, captured by what we refer to as Secondary Geometry.
For polygons, the secondary geometry is the "Associahedron" explored by Stasheff in the sixties; in the case of polytopes, the secondary geometry 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.

This is joint work with Tomasz Lukowski and Livia Ferro.