Adjacency for scattering amplitudes from the Gröbner fan

IF 5 1区 物理与天体物理 Q1 PHYSICS, PARTICLES & FIELDS Journal of High Energy Physics Pub Date : 2023-11-02 DOI:10.1007/jhep11(2023)002
L. Bossinger, J. M. Drummond, R. Glew
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Abstract

A bstract Scattering amplitudes in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n ) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras. Starting from the Plücker ideal, we describe how the polynomial cluster variables are encoded in non-prime initial ideals associated to certain maximal cones of the positive tropical fan. Following [1] we show that extending the Plücker ideal by such variables leads to a Gröbner fan with a single maximal Gröbner cone spanned by the positive tropical rays. The associated initial ideal encodes the compatibility relations among the full set of cluster variables. Thus we find that the Gröbner structure naturally encodes both the symbol alphabet and the cluster adjacency relations exhibited by scattering amplitudes without invoking the cluster algebra at all. As a potential application of these ideas we then examine the kinematic ideal associated to non-dual conformal massless scattering written in terms of spinor helicity variables. For five-particle scattering we find that the ideal can be identified with the Plücker ideal for Gr(3 , 6) and the corresponding tropical fan contains a number of non-prime ideals which encode all additional letters of the two-loop pentagon function alphabet present in various calculations of massless five-point finite remainders.
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来自Gröbner风扇的散射振幅的邻接
平面$$ \mathcal{N} $$ N = 4超Yang-Mills理论中的抽象散射振幅表现出奇点,反映了与grassmannian Gr(4, N)及其热带对应物相关的簇代数的各个方面。在这里,我们研究了这种结构的潜在起源,并检查了它们可以从底层plencker理想的Gröbner结构中恢复的程度,重点关注有限簇代数对应的Grassmannians。从plicker理想出发,我们描述了多项式聚类变量如何编码到与某些正热带扇的极大锥相关的非素数初始理想中。下面[1]我们表明,通过这些变量扩展plpl克尔理想会得到一个Gröbner风扇,该风扇具有由正热带射线跨越的单个最大Gröbner锥。关联的初始理想编码了整组簇变量之间的兼容性关系。因此,我们发现Gröbner结构自然地编码符号字母表和由散射振幅显示的簇邻接关系,而根本不调用簇代数。作为这些思想的潜在应用,我们随后研究了用旋量螺旋度变量表示的与非对偶保形无质量散射相关的运动学理想。对于五粒子散射,我们发现理想可以与Gr(3,6)的plicker理想相识别,相应的热带扇包含许多非素数理想,这些理想编码了在各种无质量五点有限余数计算中存在的双环五边形函数字母表的所有附加字母。
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
Journal of High Energy Physics
Journal of High Energy Physics PHYSICS, PARTICLES & FIELDS-
CiteScore
10.00
自引率
46.30%
发文量
2107
审稿时长
12 weeks
期刊介绍: The aim of the Journal of High Energy Physics (JHEP) is to ensure fast and efficient online publication tools to the scientific community, while keeping that community in charge of every aspect of the peer-review and publication process in order to ensure the highest quality standards in the journal. Consequently, the Advisory and Editorial Boards, composed of distinguished, active scientists in the field, jointly establish with the Scientific Director the journal''s scientific policy and ensure the scientific quality of accepted articles. JHEP presently encompasses the following areas of theoretical and experimental physics: Collider Physics Underground and Large Array Physics Quantum Field Theory Gauge Field Theories Symmetries String and Brane Theory General Relativity and Gravitation Supersymmetry Mathematical Methods of Physics Mostly Solvable Models Astroparticles Statistical Field Theories Mostly Weak Interactions Mostly Strong Interactions Quantum Field Theory (phenomenology) Strings and Branes Phenomenological Aspects of Supersymmetry Mostly Strong Interactions (phenomenology).
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