Traintrack Calabi-Yaus from Twistor Geometry

Abstract

We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in ${\mathbb{P}}^3$. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.

23 pages, 5 figures

References in corpus (14)

• Magic identities for conformal four-point integrals
• Local Integrals for Planar Scattering Amplitudes
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Uniqueness of two-loop master contours
• PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry
• A Feynman integral via higher normal functions
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods

Cited by in corpus (19)

• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• On a procedure to derive $\epsilon$-factorised differential equations beyond polylogarithms
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• The ice cone family and iterated integrals for Calabi-Yau varieties
• The elliptic double box and symbology beyond polylogarithms
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Canonical Differential Equations Beyond Genus One
• Building Bases of Loop Integrands
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Prescriptive Unitarity with Elliptic Leading Singularities
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• An Infinite Family of Elliptic Ladder Integrals
• Cuts and Isogenies
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• The Feyn-Structure of Yangian Symmetry
• Non-Local Symmetries of Planar Feynman Integrals
• The spectrum of Feynman-integral geometries at two loops
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions
• A Feynman integral depending on two elliptic curves