论文标题
舒伯特的问题，积极性和符号字母
Schubert Problems, Positivity and Symbol Letters
论文作者
论文摘要
我们提出了一种几何方法，以生成平面$ \ mathcal {n} = 4 $ Super Yang-Mills理论的幅度/积分的符号字母，称为{\ it Schubert Qualdies}。从一环积分开始，我们发现，一旦外部运动学$ \ mathbf {z} $，在给定的线路上总是订购了动量扭曲空间中的线的相交。值得注意的是，这些有序的交叉点在一条线上的交叉比例，该线路现在保证为正，与相应的feynman积分的符号字母很好地吻合，然后直接从这种几何配置中直接结论其阳性。特别是，我们从这种方法中复制了$ 18 $乘法独立代数字母$ n = 8 $振幅最多三个循环。最后，我们将讨论概括为两循环舒伯特问题，并再次从一条线上的有序点产生了一种新的代数字母，这些字母将两个不同的平方根混合在一起。它们最近在带有$ n \ geq9 $的两环双框积分的字母中发现，预计它们将以$ k+\ ell \ geq4 $的形式出现。
We propose a geometrical approach to generate symbol letters of amplitudes/integrals in planar $\mathcal{N}=4$ Super Yang-Mills theory, known as {\it Schubert problems}. Beginning with one-loop integrals, we find that intersections of lines in momentum twistor space are always ordered on a given line, once the external kinematics $\mathbf{Z}$ is in the positive region $G_+(4,n)$. Remarkably, cross-ratios of these ordered intersections on a line, which are guaranteed to be positive now, nicely coincide with symbol letters of corresponding Feynman integrals, whose positivity is then concluded directly from such geometrical configurations. In particular, we reproduce from this approach the $18$ multiplicative independent algebraic letters for $n=8$ amplitudes up to three loops. Finally, we generalize the discussion to two-loop Schubert problems and, again from ordered points on a line, generate a new kind of algebraic letters which mix two distinct square roots together. They have been found recently in the alphabet of two-loop double-box integral with $n\geq9$, and they are expected to appear in amplitudes at $k+\ell\geq4$.
