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 (3)

• 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