Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

Abstract

In this note we revisit the maximal-codimension residues, or leading singularities, of four-dimensional L-loop traintrack integrals with massive legs, both in Feynman parameter space and in momentum (twistor) space. We identify a class of “half traintracks” as the most general degenerations of traintracks with conventional (0-form) leading singularities, although the integrals themselves still have rigidity L−12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\lfloor \\frac{L-1}{2}\\right\\rfloor $$\\end{document} due to lower-loop “full traintrack” subtopologies. As a warm-up exercise, we derive closed-form expressions for their leading singularities both via (Cauchy’s) residues in Feynman parameters, and more geometrically using the so-called Schubert problems in momentum twistor space. For L-loop full traintracks, we compute their leading singularities as integrals of (L−1)-forms, which proves that the rigidity is L−1 as expected; the form is given by an inverse square root of an irreducible polynomial quartic with respect to each variable, which characterizes an (L−1)-dim Calabi-Yau manifold (elliptic curve, K3 surface, etc.) for any L. We also briefly comment on the implications for the “symbology” of these traintrack integrals.