舒伯特问题，积极性和符号字母
Schubert Problems, Positivity and Symbol Letters
摘要：
提出了一种几何方法来生成平面$\mathcal{N}=4$ Super Yang-Mills理论中振幅/积分的符号字母，称为{ \it Schubert问题}.从单圈积分开始，我们发现，一旦外运动学$\mathbf{Z}$处于正区域$G_+(4,n)$，动量扭量空间中直线的交点总是在给定直线上有序的.值得注意的是，这些有序交点在直线上的交比，现在保证是正的，与相应的费曼积分的符号字母很好地吻合，然后直接从这样的几何配置中得出其正性。特别是，我们从这种方法再现的$18$乘法独立的代数字母的$n=8$振幅高达三个循环。最后，我们将讨论推广到双循环舒伯特问题，并再次从线上的有序点，产生一种新的代数字母混合两个不同的平方根在一起。最近在$n\geq9$的双环双箱积分的字母表中发现了它们，预计它们将在$k+\ell\geq4$的振幅中出现。
英文摘要：
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 $k+\ell\geq4$.
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