Introduction
Introduction
Boundaries of the Hyper-simplex Δn,k with amplituhedronBoundaries
Boundaries of the Hyper-simplex with amplituhedronBoundaries
Δ
n,k
The following notebook uses amplituhedronBoundaries.m which is available on GitHub https://github.com/mrmrob003/amplituhedronBoundaries/. Documentation for this package can be found online in the arXiv submission arXiv preprint arXiv:2002.07146. The package was written using Mathematica 11, and is not yet fully compatible with Mathematica 12+.
30 March, 2020 – Tomasz Łukowski and Robert Moerman
In the last decade, a revolutionary geometric picture has emerged for the description of scattering amplitudes in quantum field theory (QFT). QFT is a well-established mathematical framework for describing physical processes which take place at distances much smaller than the size of atoms. This new geometric picture is encoded in terms of positive geometries, novel geometric spaces which have only recently been introduced in mathematics. These geometries generalize the notion of simplices and convex polytopes, and their properties, closely related to the notion of “volume”, provide an alternative and more efficient method for calculating scattering amplitudes in certain quantum theories. Being polytope-like, positive geometries are fascinating objects of study in their own right with rich combinatorial structures. Last month, the authors of this article released a Mathematica package, called amplituhedronBoundaries.m momentum amplituhedron
(Łukowski and Moerman, 2020)
, for exploring three important positive geometries: the amplituhedron, the momentum amplituhedron, and the hyper-simplex. The first two geometries describe physics in the so-called planar maximally supersymmetric Yang-Mills theory, a close cousin of Quantum Chromodynamics — the theory constantly tested in particle accelerators around the world. On the other hand, the hyper-simplex is a well-studied polytope appearing in many contexts in mathematics for which, just a month ago, a remarkable connection was discovered relating it to the m=2
(Łukowski, Parisi and Williams, 2020)
.The package provides important tools for generating and studying the combinatorics of the boundaries of these positive geometries and for representing this information graphically.Loading amplituhedonBoundaries.m
In[1]:=
SetDirectory[NotebookDirectory[]];<<amplituhedronBoundaries`
In order to illustrate some of the functionality of this package, we shall introduce and explore one of the aforementioned positive geometries: the hyper-simplex. All functions which we describe here have equivalent avatars for the amplituhedron and the momentum amplituhedron. The latter geometries, however, are much more complicated to describe as they rely on more sophisticated mathematical notions. Nevertheless, we will be able to illustrate the usefulness of the package already in this simpler case. We have tried to make this article as self-contained as possible, with references to other websites and articles included as optional reading.
Definitions
Definitions
The Standard SimplexThe standard simplex is the -dimensional polytope spanned by the convex hull of the standard unit vectors in :(1) ≡(,…,)∈:=1and≥0.Well-known examples of simplices include• – a point• – a line segment• – a triangle• – a tetrahedronto name a few. For more information on simplices, see
Δ
n-1
(n-1)
n
n
Δ
n-1
t
1
t
n
n
n
∑
i=1
t
i
t
i
Δ
0
Δ
1
Δ
2
Δ
3
(
Simplex
, 2020)
.For each simplex, we have a well-defined notion of boundary. For example, lets take the tetrahedron .
Δ
3
In[2]:=
tetrahedronVertices={{1,0,0},{0,1,0},{0,0,1},{1,1,1}};Graphics3D[Tetrahedron[tetrahedronVertices],ImageSize100,BaseStyleOpacity[0.5]]
Out[3]=
The tetrahedron is a -dimensional polytope. It has 4 triangular faces (2-dimensional boundaries)
3
In[4]:=
Graphics3D[#,ImageSize100]&/@Triangle/@Subsets[tetrahedronVertices,{3}]
Out[4]=
,
,
,
each triangular face, has 3 edges (1-dimensional boundaries)
In[5]:=
Graphics3D[#,ImageSize100,PlotRange{{0,1},{0,1},{0,1}},BaseStyleThick]&/@Line/@Subsets[tetrahedronVertices,{2}]
Out[5]=
,
,
,
,
,
and it too is Eulerian.
Outlook
Outlook
References
References