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−1 2 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.

Feynman integrals, geometries and differential equations

“…The latter case is already required for rather simple Feynman integrals, as for example the family of banana graphs. In this talk we put an emphasis on Calabi-Yau geometries [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], which are generalisations of elliptic curves (one-dimensional varieties) to higher dimensions.…”

Motivic geometry of two-loop Feynman integrals

“…Throughout the same period, physicists have continued to study Feynman integrals in a more direct manner. It has become clear that, in certain families of examples with few edges and vertices, the resulting Feynman integrals tend to be composed of a limited collection of building blocks including multiple polylogarithms [9], elliptic functions, elliptic polylogarithms [14,18], and more generally motivic periods of (singular) Calabi-Yau varieties [6,7,10,11,12,13,15,16,23,24,29,31,39,47]. This suggests that the periods attached to the graphs studied in the works listed above in this paragraph are, up to mixed Tate factors, related to elliptic curves and Calabi-Yau varieties.…”