Symbol alphabets from plabic graphs III: n=9

Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. In this paper we solve the CZ = 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 G(+)(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.

Automated summary

In this paper, we study the symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory, solving the CZ = 0 matrix equations associated with the totally non-negative Grassmannian using methods from previous works. We identify sets of parameterizations that produce solutions for specific cells and determine the origins of algebraic letters through plabic graphs.