Theoretical Physics Wednesday Seminar: Matteo Parisi
- Date: –15:00
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 Å4101
- Lecturer: Matteo Parisi
- Organiser: Vladimir Procházka
- Contact person: Vladimir Procházka
Amplituhedra: from Geometry to Scattering Amplitudes
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. These are extracted from a differential form, the canonical form of the Amplituhedron, which emerges from a purely geometric definition.
I will give a gentle introduction to the (tree) amplituhedra, describing them in various special cases, and explaining their relation with scattering amplitudes.
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 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 new geometric directions in my works on the Amplituhedron in momentum space and on the geometric origin of the cluster phenomena in scattering amplitudes.