Conformal geometry of null hexagons for Wilson loops and scattering amplitudes

TL;DR

This paper characterizes all conformal classes of null hexagon configurations in Minkowski space, crucial for understanding Wilson loops and scattering amplitudes, by analyzing conformal equivalence beyond cross-ratios.

Contribution

It provides a comprehensive classification of null hexagon conformal classes, including boundary cases involving infinity, advancing the geometric understanding relevant for gauge theory calculations.

Findings

Classified conformal equivalence classes of null hexagons.

Extended analysis to configurations closed via infinity.

Clarified the role of conformal transformations in these configurations.

Abstract

The cross-ratios do not uniquely fix the class of conformally equivalent configurations of null polygons. In view of applications to Wilson loops and scattering amplitudes we characterise all conformal classes of null hexagon configurations belonging to given points in cross-ratio space. At first this is done for the ordered set of vertices. Including the edges, we then investigate the equivalence classes under conformal transformations for null hexagons. This is done both for the set of null hexagons closed in finite domains of Minkowski space as well as for the set including those closed via infinity.