Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

Abstract

A bstract 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 $$ \left\lfloor \frac{L-1}{2}\right\rfloor $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> 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.

Topics & Concepts