A Note on Letters of Yangian Invariants

TL;DR

This paper explores the algebraic structure of Yangian invariants in planar ${ m ext{N}=4}$ SYM, linking their 'letters' to cluster algebras and identifying algebraic letters relevant for two-loop amplitudes.

Contribution

It introduces an algebraic approach to determine the 'letters' of Yangian invariants and connects them to cluster algebras of Grassmannian $G(4,n)$, including algebraic letters for specific amplitude configurations.

Findings

Letters of Yangian invariants relate to cluster ${ m ext{A}}$ coordinates of $G(4,n)$.

For $n=6,7$, all Yangian invariants' letters include cluster ${ m ext{A}}$ coordinates.

Algebraic letters for four-mass boxes at $n=8$ match recent two-loop amplitude symbol letters.

Abstract

Motivated by reformulating Yangian invariants in planar ${\cal N}=4$ SYM directly as $d\log$ forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the $d\log$'s, which we call "letters", for any Yangian invariant. These are functions of momentum twistors $Z$'s, given by the positive coordinates $\alpha$'s of parametrizations of the matrix $C(\alpha)$, evaluated on the support of polynomial equations $C(\alpha) \cdot Z=0$. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian $G(4,n)$, which is relevant for the symbol alphabet of $n$-point scattering amplitudes. For $n=6,7$, the collection of letters for all Yangian invariants contains the cluster ${\cal A}$ coordinates of $G(4,n)$. We determine algebraic letters of Yangian invariant associated with any "four-mass" box, which for $n=8$ reproduce the $18$ multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.