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On the Boundaries of the m=2 Amplituhedron

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On the Boundaries of the m=2 Amplituhedron. / Lukowski, Tomasz.

In: ANNALES DE L’INSTITUT HENRI POINCARÉ D, 15.02.2020.

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@article{5c33dd287a424c1f9b938e305ff1dcea,
title = "On the Boundaries of the m=2 Amplituhedron",
abstract = "Amplituhedra A_{n,k}^{(m)} are geometric objects of great interest in modern mathematics and physics: for mathematicians they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron A^{(4)}_{n,k} encodes the scattering amplitudes of the planar N=4 super Yang-Mills theory. In this paper we study the structure of boundaries for the amplituhedron A_{n,k}^{(2)}. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for m=2 is homeomorphic to a closed ball. ",
keywords = "hep-th, math.CO",
author = "Tomasz Lukowski",
year = "2020",
month = feb,
day = "15",
language = "English",
journal = "ANNALES DE L{\textquoteright}INSTITUT HENRI POINCAR{\'E} D",
issn = "2308-5827",
publisher = "European Mathematical Society Publishing House",

}

RIS

TY - JOUR

T1 - On the Boundaries of the m=2 Amplituhedron

AU - Lukowski, Tomasz

PY - 2020/2/15

Y1 - 2020/2/15

N2 - Amplituhedra A_{n,k}^{(m)} are geometric objects of great interest in modern mathematics and physics: for mathematicians they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron A^{(4)}_{n,k} encodes the scattering amplitudes of the planar N=4 super Yang-Mills theory. In this paper we study the structure of boundaries for the amplituhedron A_{n,k}^{(2)}. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for m=2 is homeomorphic to a closed ball.

AB - Amplituhedra A_{n,k}^{(m)} are geometric objects of great interest in modern mathematics and physics: for mathematicians they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron A^{(4)}_{n,k} encodes the scattering amplitudes of the planar N=4 super Yang-Mills theory. In this paper we study the structure of boundaries for the amplituhedron A_{n,k}^{(2)}. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for m=2 is homeomorphic to a closed ball.

KW - hep-th

KW - math.CO

M3 - Article

JO - ANNALES DE L’INSTITUT HENRI POINCARÉ D

JF - ANNALES DE L’INSTITUT HENRI POINCARÉ D

SN - 2308-5827

ER -