Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators

TL;DR

This paper explores the combinatorial and geometric structures of Wilson loop diagrams in SYM N=4 theory, focusing on positroids, Grassmann necklaces, and integrand denominators, providing algorithms and proofs for their properties.

Contribution

It introduces an algorithm to derive Grassmann necklaces from Wilson loop diagrams and proves the dimension relation of associated cells.

Findings

Derived Grassmann necklaces directly from diagrams

Proved cell dimension is thrice the number of propagators

Linked integrand denominators to Grassmann necklace minors

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. In this paper we study the structure of the associated positroids, as well as the structure of the denominator of the integrand defined by each diagram. We give an algorithm to derive the Grassmann necklace of the associated positroid directly from the Wilson loop diagram, and a recursive proof that the dimension of these cells is thrice the number of propagators in the diagram. We also show that the ideal generated by the denominator in the integrand is the radical of the ideal generated by the product of Grassmann necklace minors.