# Amplituhedron

The amplituhedron ia a mathematically-defined geometric structure that theoretical physicists use to simplify calculations of particle interactions in some quantum field theories. It was first published in 2013 by Nima Arkani-Hamed and Jaroslav Trnka.

## Quotes

* It has been recently conjectured that scattering amplitudes in planar N = 4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N$^k$MHV amplitudes.
** Livia Ferro, Tomasz Łukowski, Andrea Orta, and Matteo Parisi. (2016). "Towards the amplituhedron volume". Journal of High Energy Physics (3): 14–34. ISSN 1029-8479. DOI:10.1007/JHEP03(2016)014. NMHV is an acronym for "next-to-maximally-helicity-violating" — see MHV amplitudes. (quoted abstract of preprint)
* In the past decade a new, geometric picture has emerged for scattering amplitudes in planar N = 4 super Yang-Mills (SYM) theory. It originated from the observation that the tree-level amplitudes and loop-level integrands of n-point amplitudes for all helicity sectors can be computed using integrals over the Grassmannian space ... In such formulation, amplitudes can be extracted from a Grassmannian integral over a suitable contour which selects a particular sum of residues. Building upon this idea, novel studies revealed the interrelation between the rich combinatorial structure of positive Grassmannians and the physical properties of amplitudes ... From this point of view, the aforementioned residues are associated with positroid cells, which are particular subvarieties inside the positive Grassmannian.
** David Damgaard, Livia Ferro, Tomasz Łukowski, and Matteo Parisi: (2019). "The momentum amplituhedron". Journal of High Energy Physics 2019 (8). DOI:10.1007/JHEP08(2019)042. preprint (p. 1 of preprint)

### The Amplituhedron (2013)

* Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point. In this note we provide such an understanding for N = 4 SYM scattering amplitudes in the planar limit, which we identify as “the volume” of a new mathematical object–the Amplituhedron–generalizing the positive Grassmannian. Locality and unitarity emerge hand-in-hand from positive geometry.
** Nima Arkani-Hamed and Jaroslav Trnka: (2014). "The Amplituhedron". Journal of High Energy Physics 2014 (10). DOI:10.1007/JHEP10(2014)030. 2013 preprint (quoted abstract)
* The amplituhedron has now given us a new description of planar N = 4 SYM amplitudes which does not have a usual space-time/quantum mechanical description, but does make all the symmetries manifest. This is not a “duality” in the usual sense, since we are not identifying an equivalence between existing theories with familiar physical interpretations. We are seeing something rather different: new mathematical structures for representing the physics without reference to standard physical ideas, but with all symmetries manifest. Might there be an analogous story for superstring scattering amplitudes?
** 2013 preprint, p. 31