Symbol Alphabets from Plabic Graphs III: n=9

TL;DR

This paper advances the understanding of symbol alphabets in N=4 super-Yang-Mills theory by solving matrix equations related to the Grassmannian, identifying all relevant cluster and algebraic letters for the 2-loop nine-particle NMHV amplitude.

Contribution

It provides a complete characterization of the symbol alphabet for the 2-loop nine-particle NMHV amplitude using plabic graphs and Grassmannian methods.

Findings

Identified all cluster variable letters for the amplitude.

Determined the algebraic letters originating from specific plabic graphs.

Solved matrix equations for cells of the non-negative Grassmannian.

Abstract

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix equations associated to several cells of the totally non-negative Grassmannian, combining methods of arXiv:2012.15812 for rational letters and arXiv:2007.00646 for algebraic letters. We identify sets of parameterizations of the top cell of Gr_+(5,9) for which the solutions produce all of (and only) the cluster variable letters of the 2-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate.