On the Boundaries of the m=2 Amplituhedron

Tomasz Lukowski

doi:10.4171/AIHPD/124

On the Boundaries of the m=2 Amplituhedron

Tomasz Lukowski

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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.

Original language	English
Pages (from-to)	525–541
Number of pages	17
Journal	Annales de l’Institut Henri Poincaré D (AIHPD)
Volume	9
Issue number	3
DOIs	https://doi.org/10.4171/AIHPD/124
Publication status	Published - 23 Dec 2022

Keywords

hep-th
math.CO

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© 2022 Association Publications de l’Institut Henri Poincaré. Published by EMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY), https://creativecommons.org/licenses/by/4.0/
Final published version, 448 KBLicence: CC BY

Cite this

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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. ",

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On the Boundaries of the m=2 Amplituhedron

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