Rado matroids and a graphical calculus for boundaries of Wilson loop diagrams

TL;DR

This paper introduces a new diagrammatic approach to study boundaries of positroid cells from N=4 super Yang Mills theory, generalizing Wilson loop diagrams, with an algorithm and graphical calculus to analyze their structure.

Contribution

It develops a novel graphical calculus for noncrossing generalized Wilson loop diagrams, enabling direct analysis of positroid boundaries without cryptomorphisms, and provides a Python implementation.

Findings

Conditions for generalized Wilson loop diagrams to correspond to positroids.

An explicit algorithm to compute the Grassmann necklace of these positroids.

Demonstration that diagrammatic moves generate boundaries of positroids in certain cases.

Abstract

We study the boundaries of the positroid cells which arise from N = 4 super Yang Mills theory. Our main tool is a new diagrammatic object which generalizes the Wilson loop diagrams used to represent interactions in the theory. We prove conditions under which these new generalized Wilson loop diagrams correspond to positroids and give an explicit algorithm to calculate the Grassmann necklace of said positroids. Then we develop a graphical calculus operating directly on noncrossing generalized Wilson loop diagrams. In this paradigm, applying diagrammatic moves to a generalized Wilson loop diagram results in new diagrams that represent boundaries of its associated positroid, without passing through cryptomorphisms. We provide a Python implementation of the graphical calculus and use it to show that the boundaries of positroids associated to ordinary Wilson loop diagram are generated by our diagrammatic moves in certain cases.