The amplituhedron: algebra, combinatorics, and physics
Thu, April 8, 2021
Computing amplitudes is the central objective in high-energy physics.
For decades, the procedure of summing over Feynman diagrams in
perturbative quantum field theory was the name of the game.
However, performing this task in practice is severely hampered by the
combinatorial explosion of the number of these diagrams.
In 2013, N. Arkani-Hamed and J. Trnka introduced the amplituhedron,
which proposes a fruitful geometric-combinatorial approach to
computing (tree-level) amplitudes.
Relating them to volumes of certain polytopes in the realm of Grassmannians
opens up the possibility of a severe reduction in computational complexity.
This 1-day series of seminar talks aims to make the underlying
mathematics (and the bridge to physics) transparent for an interested
audience.
Sessions will be hosted via Zoom and/or gather.town.
Registration is now closed. Please send us an email if you want to join last-minute.
Abstracts
Title: Introduction to the amplituhedron
Abstract: The amplituhedron was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for calculating scattering amplitudes in planar N=4 supersymmetric Yang-Mills theory. It is defined as a projection of the totally nonnegative part of the Grassmannian, and generalizes the notion of a cyclic polytope into the Grassmannian. I will introduce the Grassmannian and the amplituhedron, and survey several results and open problems. I will touch on topics including triangulations, the BCFW recursion, topology, positive geometries, canonical forms, and duality.
Title: Eulerian numbers and the m=2 amplituhedron: signs and
triangulations from the hypersimplex
Abstract:
The hypersimplex Delta_{k+1,n} is the image of the positive Grassmannian Gr+(k+1,n) under the moment map. It is a polytope of dimension n-1 which lies in R^n. Meanwhile, the m=2 amplituhedron A_{n,k,2} is the projection of the positive Grassmannian Gr+(k,n) into the Grassmannian Gr(k,k+2). It is not a polytope, and has full dimension 2k inside Gr(k,k+2). Nevertheless, these two objects appear to be closely related. The hypersimplex can be decomposed into simplices which are counted by the Eulerian numbers. The m=2 amplituhedron can be decomposed into "chambers" which are counted by the same Eulerian numbers. Positroid triangulations of the hypersimplex are in bijection with positroid triangulations of the amplituhedron. I will describe these and other striking parallels, discovered in joint works with Lukowski-Parisi, and Parisi-Sherman-Bennett.
Title: Amplituhedra for biadjoint scalar phi^3 theory
Abstract: In this talk, I will explain some of the features of the amplituhedron story as it applies to biadjoint scalar phi^3 theory. The central insight, which is due to Arkani-Hamed, Bai, He, and Yan (arXiv:1711.09102), is that in this case the object that plays the rôle of the tree-level amplituhedron is a familiar polytope: the associahedron, originally introduced by Stasheff in homotopy theory. One of the instructive features of this story (unlike that for N=4 super Yang-Mills) is that one can see simultaneously the Feynman-diagrammatic and the geometric (amplituhedron-style) approaches. I will also say something about higher loop-level amplituhedra for this theory, and the connection to cluster algebras. (This connection starts at tree level: the 1-skeleton of the associahedron can be viewed as the exchange graph of a type A cluster algebra.)
Title: Positive geometries
Abstract: Positive geometries are semialgebraic sets equipped with
distinguished meromorphic volume forms, called canonical forms.
Examples include polytopes, positive parts of toric varieties and
Grassmannians, and conjecturally, the amplituhedron. I will survey
the subject, and if time permits, discuss a ``stringy" deformation of
canonical forms.
