Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

TL;DR

This paper explores the combinatorial and geometric structures underlying Wilson loop diagrams in SYM N=4 theory, linking them to matroids, polytopes, and associahedra to classify equivalence classes and enumerate diagram-positroid correspondences.

Contribution

It characterizes when two Wilson loop diagrams yield the same positroid, relates exact subdiagrams to uniform matroids, and enumerates diagrams per positroid cell, advancing the combinatorial understanding of these diagrams.

Findings

Conditions for Wilson loop diagrams to have the same positroid

Exact subdiagrams correspond to uniform matroids

Enumeration of diagrams per positroid cell

Abstract

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) correspond to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.