A cluster of results on amplituhedron tiles

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, Lauren Williams
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引用次数: 0

Abstract

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in \(\mathcal {N}=4\) super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the \(m=4\) amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for \(\text{ Gr}_{4,n}\). Secondly, we exhibit a tiling of the \(m=4\) amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for \(\text{ Gr}_{4,n}\). This paper is a companion to our previous paper “Cluster algebras and tilings for the \(m=4\) amplituhedron.”

Abstract Image

关于振子面砖的一组成果
振型多面体是一个数学对象,它的引入是为了提供(\mathcal {N}=4\ )超杨-米尔斯理论中散射振幅的几何起源。它概括了循环多面体和正格拉斯曼,并具有与簇代数相关的非常丰富的组合学。在这篇文章中,我们提供了一系列关于 \(m=4\) 振面的瓦片和倾斜的结果。首先,我们用簇变量为 \(\text{ Gr}_{4,n}\) 提供了 BCFW 瓦片面的完整表征。其次,我们展示了一个 \(m=4\) 振面的瓦片,它涉及一个并非来自 BCFW 循环的瓦片--spurion 瓦片,它也满足所有簇属性。最后,为了加强与簇代数的联系,我们证明了每个标准 BCFW 瓦片都是簇多样性的正部分,这使得我们可以明确地根据 \(\text{ Gr}_{4,n}\) 的簇变量来计算每个这样的瓦片的规范形式。本文是我们之前的论文 "Cluster algebras and tilings for the \(m=4\) amplituhedron" 的姐妹篇。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Letters in Mathematical Physics
Letters in Mathematical Physics 物理-物理:数学物理
CiteScore
2.40
自引率
8.30%
发文量
111
审稿时长
3 months
期刊介绍: The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.
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