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.

refs updated; 36 pages, 12 figures

References in corpus (19)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• The kite integral to all orders in terms of elliptic polylogarithms
• Elliptic polylogarithms and Feynman parameter integrals
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Cluster algebras for Feynman integrals
• Maximal Cuts in Arbitrary Dimension
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• Exponential suppression with four legs and an infinity of loops
• Manifestly Dual-Conformal Loop Integration
• Traintrack Calabi-Yaus from Twistor Geometry
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Schubert Problems, Positivity and Symbol Letters
• Modular transformations of elliptic Feynman integrals
• A study of Feynman integrals with uniform transcendental weights and the symbology from dual conformal symmetry
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills

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