Wilson Loop diagrams and Positroids

TL;DR

This paper explores the application of positive Grassmannians to Wilson loop diagrams in N=4 SYM, introducing a matroid-based framework that reveals new algebraic structures, including non-planar diagrams defining positive Grassmannians.

Contribution

It is the first to connect Wilson loop diagrams with positive Grassmannians via matroid theory, expanding the algebraic understanding of these diagrams beyond planar cases.

Findings

Certain non-planar Wilson loop diagrams define positive Grassmannians

Matroid theory effectively encodes Wilson loop diagrams

New algebraic tools for studying Wilson loops in N=4 SYM

Abstract

In this paper, we study a new application of the positive Grassmanian to Wilson loop diagrams (or MHV diagrams) for scattering amplitudes in N=4 Super Yang-Mill theory ($N=4$ SYM). There has been much interest in studying this theory via the positive Grassmanians using BCFW recursion. This is the first attempt to study MHV diagrams for planar Wilson loop calculations (or planar amplitudes) in terms of positive Grassmannians. We codify Wilson loop diagrams completely in terms of matroids. This allows us to apply the combinatorial tools in matroid theory used to identify positroids, (non-negative Grassmannians), to Wilson loop diagrams. In doing so, we find that certain non-planar Wilson loop diagrams define positive Grassmannians. While non-planar diagrams do not have physical meaning, this finding suggests that they may have value as an algebraic tool, and deserve further investigation.