Traintrack Calabi-Yaus from Twistor Geometry

TL;DR

This paper explores the geometric structures of traintrack integrals in momentum twistor space, revealing complex algebraic varieties like genus one curves and K3 surfaces at different loop levels, advancing understanding of their mathematical properties.

Contribution

It introduces a geometric framework for analyzing traintrack integrals using twistor space, identifying new algebraic varieties at two and three loops.

Findings

Leading singularity locus at two loops is a genus one curve.

At three loops, the locus is a K3 surface.

The paper discusses properties of these varieties and their higher-loop generalizations.

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.