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 $\lfloor\frac{L-1}2\rfloor$ 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.