论文标题
部分可观测时空混沌系统的无模型预测
Adjacency for scattering amplitudes from the Gröbner fan
论文作者
论文摘要
平面$ \ Mathcal {n} = 4 $ Super Yang-Mills理论中的散射幅度表现出奇异性，反映了与Grassmannians $ {\ rm gr}（4，N）$及其热带对手相关的集群代数的各个方面。在这里，我们研究了这种结构的潜在起源，并研究了它们可以从基础plücker理想的gröbner结构中恢复的程度，重点是与有限群集代数相对应的草个植物。 从Plücker理想开始，我们描述了如何用与阳性热带风扇的某些最大锥相关的非优势初始理想编码多项式群集变量。在[1]之后，我们表明，通过这种变量扩展Plücker理想会导致一个Gröbner风扇，其最大Gröbner锥由正热带射线跨越。相关的初始理想编码整个集群变量之间的兼容性关系。因此，我们发现Gröbner结构自然地编码了通过散射幅度而无需调用群集代数而显示的符号字母和簇邻接关系。 作为这些想法的潜在应用，我们然后研究了与旋转螺旋变量一起编写的与非双重共形无质量散射相关的运动学理想。对于五粒子散射，我们发现可以使用$ {\ rm gr}（3,6）$的plücker识别理想，而相应的热带风扇包含许多非prime理想，这些理想编码了无质量加拿大式函数字母的所有其他字母，这些字母在无质量的五点五点有限量的各种计算中都存在。
Scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians ${\rm Gr}(4,n)$ 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 ${\rm Gr}(3,6)$ 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.
