Schubert Problems, Positivity and Symbol Letters

Abstract: 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 $k+\ell\geq4$.