A study in $\mathbb{G}_{\mathbb{R}, \geq 0}$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

TL;DR

This paper explores the geometric structure of Wilson loop diagrams in supersymmetric gauge theory, revealing patterns and confirming conjectures about singularity cancellations using tools from total positivity and Grassmannian geometry.

Contribution

It systematically computes positroid cells for Wilson loop diagrams in SYM N=4 and verifies the cancellation of spurious singularities in a specific multi-propagator case.

Findings

Computed positroid cells for all diagrams of a given size.

Found unexpected patterns and relationships among diagrams.

Confirmed conjecture about singularity cancellations in the studied case.

Abstract

We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in the gauge theory of Supersymmetric Yang Mills (SYM) $N=4$. By applying the tools developed to study total positivity in the real Grassmannian, we are able to systematically compute with all Wilson loop diagrams of a given size and find unexpected patterns and relationships between them. We focus on the smallest nontrivial multi-propagator case, consisting of 2 propagators on 6 vertices, and compute the positroid cells associated to each diagram and the homology of the subcomplex they generate in $\mathbb{G}_{\mathbb{R}, \geq 0}$. We also verify in this case the conjecture that the spurious singularities of the volume functional {\em do} all cancel on the codimension 1 boundaries of these cells.